{"id":112,"date":"2026-10-14T15:14:21","date_gmt":"2026-10-14T15:14:21","guid":{"rendered":"https:\/\/yourwaytosat.com\/?page_id=112"},"modified":"2026-10-16T10:14:11","modified_gmt":"2026-10-16T10:14:11","slug":"sat-learning-home","status":"publish","type":"page","link":"https:\/\/yourwaytosat.com\/","title":{"rendered":"Student login and Grade 11"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"112\" class=\"elementor elementor-112\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-10ce803 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"10ce803\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1ef41d1\" data-id=\"1ef41d1\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1d07d2e elementor-widget elementor-widget-smh-platform\" data-id=\"1d07d2e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"smh-platform.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div id=\"smh-native-1\" class=\"smh-native-platform smh-portal-page smh-teacher-skin\"><script type=\"application\/json\" class=\"smh-editor-config\">{\"restBase\":\"https:\\\/\\\/yourwaytosat.com\\\/?rest_route=\\\/sat-math\\\/v1\\\/\",\"assetBase\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\",\"teacherUrl\":\"https:\\\/\\\/yourwaytosat.com\\\/sat-teacher\\\/\",\"adminUrl\":\"https:\\\/\\\/yourwaytosat.com\\\/sat-admin\\\/\",\"grade10ProgressUrl\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-admin\\\/admin-ajax.php?action=smh_grade10_progress\",\"grade10EntryUrl\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-admin\\\/admin-post.php\",\"grade10Url\":\"https:\\\/\\\/yourwaytosat.com\\\/grade10\\\/?role=grade10\",\"quizUrl\":\"https:\\\/\\\/yourwaytosat.com\\\/secure-quiz\\\/\",\"homeUrl\":\"https:\\\/\\\/yourwaytosat.com\\\/\",\"i18n\":{\"$10\":\"$10\",\"$18\":\"$18\",\"$30\":\"$30\",\"$4\":\"$4\",\"$42\":\"$42\",\"$6\":\"$6\",\"(0, 1)\":\"(0, 1)\",\"(0, 6)\":\"(0, 6)\",\"(0, \\u22124)\":\"(0, \\u22124)\",\"(0, \\u22125)\":\"(0, \\u22125)\",\"(2, 5)\":\"(2, 5)\",\"(3, 0)\":\"(3, 0)\",\"(3, n)\":\"(3, n)\",\"(4, n)\":\"(4, n)\",\"(\\u22122, 3)\":\"(\\u22122, 3)\",\"(\\u22122,3)\":\"(\\u22122,3)\",\"(\\u22124, 0)\":\"(\\u22124, 0)\",\"(\\u22125, 0)\":\"(\\u22125, 0)\",\". 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What is the sale price?\",\"options\":[\"$20\",\"$55\",\"$60\",\"$65\"],\"answer\":2,\"drill\":\"20-Question Test\",\"hint\":\"A 25% discount means the buyer pays 75% of the original price.\",\"explanation\":\"The sale price is 0.75($80) = $60, choice C.\"},{\"prompt\":\"A school club collected $156 by selling 12 identical tickets. What was the price of each ticket?\",\"options\":[\"$11\",\"$12\",\"$13\",\"$14\"],\"answer\":2,\"drill\":\"20-Question Test\",\"hint\":\"Divide the total amount collected by the number of tickets.\",\"explanation\":\"$156 \\u00f7 12 = $13 per ticket, choice C.\"},{\"prompt\":\"A fitness center charges an $18 registration fee plus $7 per class. If Maya can spend at most $60, what is the greatest number of classes she can take?\",\"options\":[\"5\",\"6\",\"7\",\"8\"],\"answer\":1,\"drill\":\"20-Question Test\",\"hint\":\"Solve 18 + 7c \\u2264 60 and use a whole number of classes.\",\"explanation\":\"7c \\u2264 42, so c \\u2264 6. The greatest possible number is 6 classes, choice B.\"},{\"prompt\":\"A rectangle has area 96 square units. Its length is 4 units greater than its width. What is the rectangle\\u0027s length?\",\"options\":[\"8\",\"10\",\"12\",\"16\"],\"answer\":2,\"drill\":\"20-Question Test\",\"hint\":\"Let the width be w and solve w(w + 4) = 96.\",\"explanation\":\"w\\u00b2 + 4w \\u2212 96 = (w + 12)(w \\u2212 8) = 0. The positive width is 8, so the length is 12, choice C.\"},{\"prompt\":\"If 4^(x \\u2212 1) = 64, what is x?\",\"options\":[\"2\",\"3\",\"4\",\"5\"],\"answer\":2,\"drill\":\"20-Question Test\",\"hint\":\"Write both sides as powers of 2.\",\"explanation\":\"4^(x \\u2212 1) = 2^(2x \\u2212 2) and 64 = 2\\u2076. Thus 2x \\u2212 2 = 6, so x = 4, choice C.\"},{\"prompt\":\"A $75 backpack is discounted by 20%, then the sale price is discounted by another 10%. What is the final price?\",\"options\":[\"$48\",\"$54\",\"$60\",\"$67.50\"],\"answer\":1,\"drill\":\"20-Question Test\",\"hint\":\"Apply the two percentage multipliers successively.\",\"explanation\":\"$75(0.80)(0.90) = $54, so choice B is correct.\"},{\"prompt\":\"After a 20% decrease, a price is $72. What was the original price?\",\"options\":[\"$80\",\"$86.40\",\"$90\",\"$92\"],\"answer\":2,\"drill\":\"20-Question Test\",\"hint\":\"The new price is 80% of the original.\",\"explanation\":\"If p is the original price, 0.80p = 72. Therefore p = 72\\\/0.80 = $90, choice C.\"},{\"prompt\":\"The ratio of red marbles to blue marbles is 3:5. If there are 64 marbles total, how many are blue?\",\"options\":[\"24\",\"32\",\"40\",\"48\"],\"answer\":2,\"drill\":\"20-Question Test\",\"hint\":\"The ratio has 8 total parts.\",\"explanation\":\"Blue marbles make up 5 of 8 parts, so (5\\\/8)(64) = 40, choice C.\"},{\"prompt\":\"A line has slope 2\\\/3. What is the slope of a line perpendicular to it?\",\"options\":[\"\\u22123\\\/2\",\"\\u22122\\\/3\",\"2\\\/3\",\"3\\\/2\"],\"answer\":0,\"drill\":\"20-Question Test\",\"hint\":\"Perpendicular nonvertical lines have slopes that are negative reciprocals.\",\"explanation\":\"The negative reciprocal of 2\\\/3 is \\u22123\\\/2, choice A.\"},{\"prompt\":\"The endpoints of segment AB are A(\\u22124, 2) and B(8, 10). What is the midpoint of AB?\",\"options\":[\"(2, 4)\",\"(2, 6)\",\"(4, 6)\",\"(6, 8)\"],\"answer\":1,\"drill\":\"20-Question Test\",\"hint\":\"Average the x-coordinates and the y-coordinates separately.\",\"explanation\":\"The midpoint is ((\\u22124 + 8)\\\/2, (2 + 10)\\\/2) = (2, 6), choice B.\",\"visual\":\"segment\"},{\"prompt\":\"The circle shown has center (\\u22122, 3) and radius 5. Which equation represents the circle?\",\"options\":[\"(x \\u2212 2)\\u00b2 + (y + 3)\\u00b2 = 5\",\"(x + 2)\\u00b2 + (y \\u2212 3)\\u00b2 = 5\",\"(x \\u2212 2)\\u00b2 + (y + 3)\\u00b2 = 25\",\"(x + 2)\\u00b2 + (y \\u2212 3)\\u00b2 = 25\"],\"answer\":3,\"drill\":\"20-Question Test\",\"hint\":\"Use (x \\u2212 h)\\u00b2 + (y \\u2212 k)\\u00b2 = r\\u00b2.\",\"explanation\":\"With center (h, k) = (\\u22122, 3) and radius 5, the equation is (x + 2)\\u00b2 + (y \\u2212 3)\\u00b2 = 25, choice D.\",\"visual\":\"circle\"},{\"prompt\":\"What is the mean of the data represented by the frequency table?\",\"options\":[\"4.0\",\"4.375\",\"5.0\",\"5.375\"],\"answer\":1,\"drill\":\"20-Question Test\",\"hint\":\"Multiply each value by its frequency, add, and divide by the total frequency.\",\"explanation\":\"The weighted sum is 2(3) + 5(4) + 9(1) = 35, and the total frequency is 8. The mean is 35\\\/8 = 4.375, choice B.\",\"visual\":\"frequency-table\"},{\"prompt\":\"Of 60 students, 35 study algebra, 28 study geometry, and 15 study both. If a student who studies algebra is selected, what is the probability that the student also studies geometry?\",\"options\":[\"1\\\/4\",\"3\\\/7\",\"1\\\/2\",\"15\\\/28\"],\"answer\":1,\"drill\":\"20-Question Test\",\"hint\":\"Restrict the sample space to the 35 algebra students.\",\"explanation\":\"Among the 35 algebra students, 15 also study geometry. The conditional probability is 15\\\/35 = 3\\\/7, choice B.\"},{\"prompt\":\"In the right triangle shown, what is cos \\u03b8?\",\"options\":[\"5\\\/13\",\"12\\\/13\",\"5\\\/12\",\"12\\\/5\"],\"answer\":0,\"drill\":\"20-Question Test\",\"hint\":\"Cosine is adjacent side divided by hypotenuse.\",\"explanation\":\"Relative to \\u03b8, the adjacent side is 5 and the hypotenuse is 13. Therefore cos \\u03b8 = 5\\\/13, choice A.\",\"visual\":\"right-triangle\"},{\"prompt\":\"Two similar triangles have corresponding sides 6 and 9. Another pair of corresponding sides has lengths x + 2 and 2x \\u2212 1, respectively. What is x?\",\"options\":[\"6\",\"7\",\"8\",\"10\"],\"answer\":2,\"drill\":\"20-Question Test\",\"hint\":\"Set the ratio of the second triangle to the first equal to 9\\\/6.\",\"explanation\":\"(2x \\u2212 1)\\\/(x + 2) = 9\\\/6 = 3\\\/2. Thus 4x \\u2212 2 = 3x + 6, giving x = 8, choice C.\"},{\"prompt\":\"From point A on level ground, the angle of elevation to the top of a vertical tower is \\u03b8, where tan \\u03b8 = 3\\\/5. Point B lies on the same straight path from A to the tower and is 24 feet closer to the tower. From point B, the angle of elevation to the top is \\u03c6, where tan \\u03c6 = 1. What is the height of the tower, in feet?\",\"options\":[\"30\",\"36\",\"40\",\"48\"],\"answer\":1,\"drill\":\"20-Question Test\",\"hint\":\"Let x be the horizontal distance from A to the tower and h be its height. Write one tangent equation from each observation point.\",\"explanation\":\"From point A, h\\\/x = 3\\\/5, so h = 3x\\\/5. Point B is 24 feet closer, and tan \\u03c6 = h\\\/(x \\u2212 24) = 1, so h = x \\u2212 24. Equating the expressions gives 3x\\\/5 = x \\u2212 24. Thus 3x = 5x \\u2212 120, so x = 60 and h = 60 \\u2212 24 = 36 feet. Choice B is correct.\"}]},{\"title\":\"Plugging In\",\"questions\":[{\"prompt\":\"Given x = 7 \\u2212 18\\\/y, which equation correctly expresses y in terms of x?\",\"options\":[\"y = \\u221218\\\/(x \\u2212 7)\",\"y = \\u221218\\\/x \\u2212 7\",\"y = \\u221218\\\/x + 7\",\"y = (x \\u2212 7)\\\/18\"],\"answer\":0,\"drill\":\"Drill 1\",\"hint\":\"Subtract 7, then multiply both sides by y.\",\"explanation\":\"From x \\u2212 7 = \\u221218\\\/y, multiply by y to get y(x \\u2212 7) = \\u221218. Dividing by x \\u2212 7 gives y = \\u221218\\\/(x \\u2212 7), choice A.\"},{\"prompt\":\"For A = 20c + 15, how does the cost change if the number of cartons c increases by 3?\",\"options\":[\"Decreases by $15\",\"Increases by $3\",\"Increases by $45\",\"Increases by $60\"],\"answer\":3,\"drill\":\"Drill 1\",\"hint\":\"The coefficient of c is the cost change for each additional carton.\",\"explanation\":\"Each additional carton adds $20. An increase of 3 cartons changes the cost by 20\\u00b73 = $60, so the cost increases by $60, choice D.\"},{\"prompt\":\"The table shows four corresponding values of p and q. Which equation represents the linear relationship between p and q?\",\"options\":[\"q = \\u22123p \\u2212 6\",\"q = \\u22123p \\u2212 2\",\"q = \\u22126p \\u2212 6\",\"q = \\u22126p \\u2212 12\"],\"answer\":0,\"drill\":\"Drill 1\",\"hint\":\"Find the change in q when p increases by 1, then use the row where p = 0.\",\"explanation\":\"As p increases by 1, q decreases by 3, so the slope is \\u22123. When p = 0, q = \\u22126, giving an intercept of \\u22126. Therefore q = \\u22123p \\u2212 6, choice A.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/plugging-in\\\/redrawn\\\/pq-linear-table.svg\"},{\"prompt\":\"In the displayed system, p and q are constants. If p = q + 4, which relationship between x and y must be true?\",\"options\":[\"x = y \\u2212 2\",\"x = y + 2\",\"x = y + 4\",\"x = y + 8\"],\"answer\":1,\"drill\":\"Drill 1\",\"hint\":\"Simplify each equation to express p and q in terms of x and y.\",\"explanation\":\"The first equation gives p = 2x + 5, and the second gives q = 2y + 5. Since p = q + 4, 2x + 5 = 2y + 9. Thus 2x = 2y + 4 and x = y + 2, choice B.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/plugging-in\\\/redrawn\\\/pq-system.svg\"},{\"prompt\":\"The first term of a sequence is 16, and every term after the first is one-half of the preceding term. If p represents the nth term, which expression gives p in terms of n?\",\"options\":[\"p = (1\\\/2)(16)\\u207f\\u207b\\u00b9\",\"p = (1\\\/2)(16)\\u207f\",\"p = 16(1\\\/2)\\u207f\\u207b\\u00b9\",\"p = 16(1\\\/2)\\u207f\"],\"answer\":2,\"drill\":\"Drill 1\",\"hint\":\"Use the geometric-sequence formula a\\u2099 = a\\u2081r\\u207f\\u207b\\u00b9.\",\"explanation\":\"The first term is 16 and the common ratio is 1\\\/2. Therefore p = 16(1\\\/2)\\u207f\\u207b\\u00b9, choice C.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/plugging-in\\\/redrawn\\\/halving-sequence.svg\"},{\"prompt\":\"Which expression is equivalent to 7x \\u2212 13x\\u00b2?\",\"options\":[\"x(\\u22126x)\",\"x(7 \\u2212 13x)\",\"7x(1 \\u2212 13x)\",\"x\\u00b2(7 \\u2212 13x)\"],\"answer\":1,\"drill\":\"Drill 2\",\"hint\":\"Factor out the greatest common factor x.\",\"explanation\":\"Both terms contain x. Factoring it out gives 7x \\u2212 13x\\u00b2 = x(7 \\u2212 13x), choice B.\"},{\"prompt\":\"An isosceles triangle has equal sides s and a third side 4 feet shorter. What is its perimeter?\",\"options\":[\"(s \\u2212 4)\\\/2\",\"2s \\u2212 4\",\"3s \\u2212 4\",\"(s\\u00b2 \\u2212 4)\\\/2\"],\"answer\":2,\"drill\":\"Drill 2\",\"hint\":\"Write the three side lengths and add them.\",\"explanation\":\"The side lengths are s, s, and s \\u2212 4. Their sum is s + s + (s \\u2212 4) = 3s \\u2212 4, choice C.\"},{\"prompt\":\"Which expression is equivalent to x\\u00b2 \\u2212 7x + 6?\",\"options\":[\"(x \\u2212 1)(x \\u2212 6) \\u2212 7\",\"(x \\u2212 1)(x \\u2212 6) \\u2212 7x\",\"(x \\u2212 3)(x \\u2212 4) \\u2212 6\",\"(x \\u2212 3)(x \\u2212 4) \\u2212 6x\"],\"answer\":2,\"drill\":\"Drill 2\",\"hint\":\"Expand each product and combine the constant terms.\",\"explanation\":\"(x \\u2212 3)(x \\u2212 4) \\u2212 6 = x\\u00b2 \\u2212 7x + 12 \\u2212 6 = x\\u00b2 \\u2212 7x + 6. Therefore choice C is equivalent.\"},{\"prompt\":\"Which expression is equivalent to 4 \\u2212 3\\\/(a \\u2212 2)?\",\"options\":[\"(a \\u2212 11)\\\/(a \\u2212 2)\",\"(a \\u2212 8)\\\/(a \\u2212 2)\",\"(4a \\u2212 8)\\\/(a \\u2212 2)\",\"(4a \\u2212 11)\\\/(a \\u2212 2)\"],\"answer\":3,\"drill\":\"Drill 2\",\"hint\":\"Rewrite 4 with denominator a \\u2212 2, then combine the numerators.\",\"explanation\":\"Write 4 as 4(a \\u2212 2)\\\/(a \\u2212 2). Then 4 \\u2212 3\\\/(a \\u2212 2) = [4(a \\u2212 2) \\u2212 3]\\\/(a \\u2212 2) = (4a \\u2212 11)\\\/(a \\u2212 2), choice D.\"},{\"prompt\":\"Sphere A has one-third the radius of sphere B. What fraction of B\\u0027s volume is A\\u0027s volume?\",\"options\":[\"1\\\/36\",\"1\\\/27\",\"1\\\/9\",\"1\\\/3\"],\"answer\":1,\"drill\":\"Drill 2\",\"hint\":\"Sphere volume is proportional to the cube of the radius.\",\"explanation\":\"The radius scale factor from sphere B to sphere A is 1\\\/3. Volumes scale by the cube of this factor, so V_A\\\/V_B = (1\\\/3)\\u00b3 = 1\\\/27, choice B.\"},{\"prompt\":\"A shirt costs p dollars and the tax rate is t%. What is the total cost of n shirts, including tax?\",\"options\":[\"npt\",\"np(t + 1)\\\/100\",\"100n(p + pt)\",\"np(t + 100)\\\/100\"],\"answer\":3,\"drill\":\"Drill 3\",\"hint\":\"One shirt costs p(1 + t\\\/100) after tax; then multiply by n.\",\"explanation\":\"The cost of n shirts before tax is np. Including t% tax multiplies this by 1 + t\\\/100 = (t + 100)\\\/100. Thus the total is np(t + 100)\\\/100, choice D.\"},{\"prompt\":\"Which equation models 500 followers at week 0, tripling each week?\",\"options\":[\"f = (1\\\/3)\\u02b7\",\"f = (1\\\/3)(500)\\u02b7\",\"f = 3(500)\\u02b7\",\"f = 500(3)\\u02b7\"],\"answer\":3,\"drill\":\"Drill 3\",\"hint\":\"Use initial value times growth factor raised to the number of weeks.\",\"explanation\":\"At w = 0, the model must equal 500, and the values triple each week. Therefore f = 500(3)\\u02b7, which gives 500, 1,500, and 4,500 for weeks 0, 1, and 2. Choice D is correct.\"},{\"prompt\":\"Each bead kit makes 3 bracelets. How many kits K are needed to make 4 bracelets for each of x friends, where x is a positive multiple of 3?\",\"options\":[\"K = (3\\\/4)x\",\"K = 4x\",\"K = (4\\\/3)x\",\"K = 12x\"],\"answer\":2,\"drill\":\"Drill 3\",\"hint\":\"Find the total number of bracelets, then divide by 3 bracelets per kit.\",\"explanation\":\"The friends need 4x bracelets total. Since each kit makes 3 bracelets, K = 4x\\\/3 = (4\\\/3)x, choice C.\"},{\"prompt\":\"A delivery company uses 12 vans to deliver 310 packages, 18 vans to deliver 430 packages, and 25 vans to deliver 570 packages. Which linear function models the number of packages P delivered using v vans?\",\"options\":[\"P(v) = 20v + 70\",\"P(v) = 20v \\u2212 70\",\"P(v) = 12v + 166\",\"P(v) = 25v + 10\"],\"answer\":0,\"drill\":\"Drill 3\",\"hint\":\"Find the change in packages per additional van, then use one data pair to find the intercept.\",\"explanation\":\"From 12 to 18 vans, packages increase by 430 \\u2212 310 = 120 over 6 vans, so the slope is 20. Using (12, 310), 310 = 20(12) + b gives b = 70. Thus P(v) = 20v + 70, choice A.\"},{\"prompt\":\"A babysitter charges $22 per hour for the first 5 hours and $20 for each extra hour. Which function gives the cost for h \\u2265 5?\",\"options\":[\"g(h) = 42h + 10\",\"g(h) = 20h + 110\",\"g(h) = 20h + 22\",\"g(h) = 20h + 10\"],\"answer\":3,\"drill\":\"Drill 3\",\"hint\":\"The first five hours cost $110; add $20 for each of the h \\u2212 5 additional hours.\",\"explanation\":\"The first five hours cost 5($22) = $110. The remaining h \\u2212 5 hours cost $20 each, so g(h) = 110 + 20(h \\u2212 5) = 20h + 10, choice D.\"},{\"prompt\":\"A robotics team programs a printer to spend 7 minutes calibrating, then 4 minutes on each of the first 6 components. Later components take 3 minutes each. Which expression gives the total time T, in minutes, to complete n components when n \\u2265 6?\",\"options\":[\"T = 3n + 7\",\"T = 4n + 7\",\"T = 3n + 13\",\"T = 7n + 13\"],\"answer\":2,\"drill\":\"Drill 3\",\"hint\":\"Include calibration, the first six components, and the remaining components separately.\",\"explanation\":\"T = 7 + 6(4) + 3(n \\u2212 6) = 3n + 13. For n = 6, the expression correctly gives 31 minutes.\"},{\"prompt\":\"Given x = 7 \\u2212 18\\\/y, which equation correctly expresses y in terms of x?\",\"options\":[\"y = \\u221218\\\/(x \\u2212 7)\",\"y = \\u221218\\\/x \\u2212 7\",\"y = \\u221218\\\/x + 7\",\"y = (x \\u2212 7)\\\/18\"],\"answer\":0,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"The first term of a sequence is 16, and every term after the first is one-half of the preceding term. If p represents the nth term, which expression gives p in terms of n?\",\"options\":[\"p = (1\\\/2)(16)\\u207f\\u207b\\u00b9\",\"p = (1\\\/2)(16)\\u207f\",\"p = 16(1\\\/2)\\u207f\\u207b\\u00b9\",\"p = 16(1\\\/2)\\u207f\"],\"answer\":2,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/plugging-in\\\/redrawn\\\/halving-sequence.svg\"},{\"prompt\":\"Which expression is equivalent to 4 \\u2212 3\\\/(a \\u2212 2)?\",\"options\":[\"(a \\u2212 11)\\\/(a \\u2212 2)\",\"(a \\u2212 8)\\\/(a \\u2212 2)\",\"(4a \\u2212 8)\\\/(a \\u2212 2)\",\"(4a \\u2212 11)\\\/(a \\u2212 2)\"],\"answer\":3,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"Which equation models 500 followers at week 0, tripling each week?\",\"options\":[\"f = (1\\\/3)\\u02b7\",\"f = (1\\\/3)(500)\\u02b7\",\"f = 3(500)\\u02b7\",\"f = 500(3)\\u02b7\"],\"answer\":3,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"A robotics team programs a printer to spend 7 minutes calibrating, then 4 minutes on each of the first 6 components. Later components take 3 minutes each. Which expression gives the total time T, in minutes, to complete n components when n \\u2265 6?\",\"options\":[\"T = 3n + 7\",\"T = 4n + 7\",\"T = 3n + 13\",\"T = 7n + 13\"],\"answer\":2,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"domain\":\"Algebra\",\"prompt\":\"A student wants to plug in n = 2 to evaluate (n\\u00b2 \\u2212 4)\\\/(n \\u2212 2). What mistake would this create?\",\"options\":[\"The numerator would be negative\",\"The denominator would equal zero\",\"The value of n must be odd\",\"The expression would equal 1\"],\"answer\":1,\"drill\":\"Common Mistakes\",\"hint\":\"Check the denominator before substituting.\",\"explanation\":\"Substituting n = 2 makes the denominator 2 \\u2212 2 = 0. Division by zero is undefined, so 2 is not a valid test value.\"},{\"domain\":\"Algebra\",\"prompt\":\"Positive integers x and y satisfy x + y = 10 and xy = 21. What is the value of x\\u00b2 + y\\u00b2?\",\"options\":[\"42\",\"58\",\"79\",\"100\"],\"answer\":1,\"drill\":\"Challenge Question\",\"hint\":\"Square x + y and use xy = 21.\",\"explanation\":\"(x + y)\\u00b2 = x\\u00b2 + 2xy + y\\u00b2. Therefore, 100 = x\\u00b2 + y\\u00b2 + 42, so x\\u00b2 + y\\u00b2 = 58.\"}]},{\"title\":\"Plugging In Answers\",\"questions\":[{\"prompt\":\"A school garden receives a batch of seeds. After 12% fail to sprout, the team plants 22 trays with 24 seedlings in each tray and has no seedlings left. How many seeds were in the batch?\",\"options\":[\"528\",\"576\",\"600\",\"660\"],\"answer\":2,\"drill\":\"Drill 1\",\"hint\":\"Test which starting amount leaves 22 times 24 seedlings after a 12% loss.\",\"explanation\":\"The trays contain 22(24) = 528 seedlings. These represent 88% of the original seeds, so the batch contained 528\\\/0.88 = 600 seeds.\"},{\"prompt\":\"Which ordered pair solves y = 3x + 5 and y = (x \\u2212 1)(x + 9)?\",\"options\":[\"(\\u22127, \\u221216)\",\"(\\u221216, \\u22127)\",\"(1, \\u22129)\",\"(\\u22129, 1)\"],\"answer\":0,\"drill\":\"Drill 1\",\"hint\":\"Substitute each ordered pair into both equations.\",\"explanation\":\"For (\\u22127, \\u221216), the linear equation gives 3(\\u22127) + 5 = \\u221216. The quadratic expression gives (\\u22127 \\u2212 1)(\\u22127 + 9) = (\\u22128)(2) = \\u221216. The pair satisfies both equations, so choice A is correct.\"},{\"prompt\":\"A total of 68 items is split into c and d items, where c is 10 less than twice d. What is c?\",\"options\":[\"26\",\"39\",\"42\",\"44\"],\"answer\":2,\"drill\":\"Drill 1\",\"hint\":\"Use c + d = 68 and c = 2d \\u2212 10.\",\"explanation\":\"Substitute c = 2d \\u2212 10 into c + d = 68: 2d \\u2212 10 + d = 68, so 3d = 78 and d = 26. Then c = 2(26) \\u2212 10 = 42, choice C.\"},{\"prompt\":\"A circle\\u0027s circumference is split into two arcs totaling 24 inches. The smaller arc is one-third the larger. What is the smaller arc\\u0027s length?\",\"options\":[\"3\",\"6\",\"8\",\"18\"],\"answer\":1,\"drill\":\"Drill 1\",\"hint\":\"Let the smaller arc be s; then the larger arc is 3s.\",\"explanation\":\"The two arc lengths are s and 3s, and together they equal 24. Thus 4s = 24, so s = 6 inches, choice B.\"},{\"prompt\":\"An order has 20 bracelets. Small bracelets use 5 beads, large bracelets use 8, and 133 beads are used total. How many are large?\",\"options\":[\"7\",\"9\",\"10\",\"11\"],\"answer\":3,\"drill\":\"Drill 1\",\"hint\":\"If \\u2113 bracelets are large, then 20 \\u2212 \\u2113 are small.\",\"explanation\":\"Let \\u2113 be the number of large bracelets. Then 5(20 \\u2212 \\u2113) + 8\\u2113 = 133. Simplifying gives 100 + 3\\u2113 = 133, so 3\\u2113 = 33 and \\u2113 = 11, choice D.\"},{\"prompt\":\"What is the solution (x, y) to 2x + 6y = 0 and x \\u2212 y = 4?\",\"options\":[\"(\\u22123, \\u22121)\",\"(\\u22121, 3)\",\"(3, \\u22121)\",\"(6, \\u22121)\"],\"answer\":2,\"drill\":\"Drill 2\",\"hint\":\"From x \\u2212 y = 4, write x = y + 4 and substitute.\",\"explanation\":\"Substitute x = y + 4 into 2x + 6y = 0: 2(y + 4) + 6y = 0, so 8y + 8 = 0 and y = \\u22121. Then x = \\u22121 + 4 = 3. The solution is (3, \\u22121), choice C.\"},{\"prompt\":\"The system y = x\\u00b2 + 11 and y = 47 intersects at (x, y). Which could be x?\",\"options\":[\"6\",\"11\",\"36\",\"47\"],\"answer\":0,\"drill\":\"Drill 2\",\"hint\":\"Set x\\u00b2 + 11 equal to 47.\",\"explanation\":\"x\\u00b2 + 11 = 47 gives x\\u00b2 = 36, so x = 6 or x = \\u22126. Of the choices, 6 could be the x-coordinate, choice A.\"},{\"prompt\":\"What is the positive solution to x\\u00b2 \\u2212 4x \\u2212 32 = 0?\",\"options\":[\"\\u22124\",\"4\",\"8\",\"16\"],\"answer\":2,\"drill\":\"Drill 2\",\"hint\":\"Find two numbers whose product is \\u221232 and sum is \\u22124.\",\"explanation\":\"The quadratic factors as (x \\u2212 8)(x + 4) = 0. Its solutions are x = 8 and x = \\u22124, so the positive solution is 8, choice C.\"},{\"prompt\":\"If x = 2 solves (kx \\u2212 1)\\u00b2 = 25, which listed value could be k?\",\"options\":[\"\\u22126\",\"\\u22122\",\"4\",\"5\"],\"answer\":1,\"drill\":\"Drill 2\",\"hint\":\"Substitute x = 2, then take both the positive and negative square roots of 25.\",\"explanation\":\"Substituting x = 2 gives (2k \\u2212 1)\\u00b2 = 25, so 2k \\u2212 1 = 5 or 2k \\u2212 1 = \\u22125. These yield k = 3 or k = \\u22122. Of the choices, \\u22122 is possible, choice B.\"},{\"prompt\":\"Which value solves 2(x \\u2212 3)\\u00b2 + 4(x \\u2212 3) + 2 = 0?\",\"options\":[\"\\u22122\",\"0\",\"1\",\"2\"],\"answer\":3,\"drill\":\"Drill 2\",\"hint\":\"Let u = x \\u2212 3 and recognize a perfect-square trinomial.\",\"explanation\":\"Let u = x \\u2212 3. The equation becomes 2u\\u00b2 + 4u + 2 = 2(u + 1)\\u00b2 = 0, so u = \\u22121. Therefore x \\u2212 3 = \\u22121 and x = 2, choice D.\"},{\"prompt\":\"Jenny made 9 fewer cupcakes than George, and together they made 73. How many did George make?\",\"options\":[\"32\",\"41\",\"48\",\"82\"],\"answer\":1,\"drill\":\"Drill 3\",\"hint\":\"If George made g cupcakes, Jenny made g \\u2212 9.\",\"explanation\":\"g + (g \\u2212 9) = 73 gives 2g \\u2212 9 = 73, so 2g = 82 and g = 41. George made 41 cupcakes, choice B.\"},{\"prompt\":\"A photography club buys a camera and a carrying case priced at 96 AED. A 6% handling fee is charged on their combined price. The final bill is 1,310.16 AED. What is the camera price before the fee, in AED?\",\"options\":[\"1,044\",\"1,140\",\"1,236\",\"1,310\"],\"answer\":1,\"drill\":\"Drill 3\",\"hint\":\"Test each camera price after adding the case price and applying the fee.\",\"explanation\":\"The combined price before the fee is 1,310.16\\\/1.06 = 1,236 AED. Subtracting the 96 AED case gives a camera price of 1,140 AED.\"},{\"prompt\":\"A rectangular mural has area 96 square meters, and its height is 4 meters less than its width. What is its height?\",\"options\":[\"4\",\"8\",\"12\",\"24\"],\"answer\":1,\"drill\":\"Drill 3\",\"hint\":\"Look for two factors of 96 that differ by 4.\",\"explanation\":\"Let the height be h, so the width is h + 4. Then h(h + 4) = 96. The positive factor pair 8 and 12 differs by 4, so h = 8 meters, choice B.\"},{\"prompt\":\"Three-fifths of a club\\u0027s members are juniors. If 15 are juniors, how many members are not juniors?\",\"options\":[\"6\",\"9\",\"10\",\"25\"],\"answer\":2,\"drill\":\"Drill 3\",\"hint\":\"First find the total membership from (3\\\/5)T = 15.\",\"explanation\":\"If (3\\\/5)T = 15, then T = 15(5\\\/3) = 25 total members. The number who are not juniors is 25 \\u2212 15 = 10, choice C.\"},{\"prompt\":\"A group of 84 people is split into three groups. The largest has three times as many people as the two smaller groups combined. How many are in the smaller groups combined?\",\"options\":[\"21\",\"28\",\"42\",\"63\"],\"answer\":0,\"drill\":\"Drill 3\",\"hint\":\"Let s be the combined size of the two smaller groups; then the largest has 3s.\",\"explanation\":\"The total is s + 3s = 84, so 4s = 84 and s = 21. The two smaller groups contain 21 people combined, choice A.\"},{\"prompt\":\"A jacket\\u0027s sale price is 25% less than its original price, and the original price is $15 more than the sale price. What is the sale price?\",\"options\":[\"$15\",\"$30\",\"$45\",\"$60\"],\"answer\":2,\"drill\":\"Drill 3\",\"hint\":\"The $15 difference is 25% of the original price.\",\"explanation\":\"Let O be the original price. The discount is 0.25O, and it equals $15, so O = 15\\\/0.25 = $60. The sale price is 75% of $60, or $45, choice C.\"},{\"prompt\":\"A student photographer receives 175 AED for setting up an event and 28 AED for each photo package sold. An additional bonus equal to 5% of package revenue is paid. What is the least number of packages needed to earn at least 2,500 AED in total?\",\"options\":[\"78\",\"79\",\"80\",\"85\"],\"answer\":2,\"drill\":\"Drill 3\",\"hint\":\"Account for the fixed payment before testing the whole-number sales counts.\",\"explanation\":\"Total earnings for n packages are 175 + 28(1.05)n = 175 + 29.4n. At 79 packages the total is 2,497.60 AED, while at 80 it is 2,527 AED. The minimum is 80.\"},{\"prompt\":\"A school garden receives a batch of seeds. After 12% fail to sprout, the team plants 22 trays with 24 seedlings in each tray and has no seedlings left. How many seeds were in the batch?\",\"options\":[\"528\",\"576\",\"600\",\"660\"],\"answer\":2,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"An order has 20 bracelets. Small bracelets use 5 beads, large bracelets use 8, and 133 beads are used total. How many are large?\",\"options\":[\"7\",\"9\",\"10\",\"11\"],\"answer\":3,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"If x = 2 solves (kx \\u2212 1)\\u00b2 = 25, which listed value could be k?\",\"options\":[\"\\u22126\",\"\\u22122\",\"4\",\"5\"],\"answer\":1,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"A rectangular mural has area 96 square meters, and its height is 4 meters less than its width. What is its height?\",\"options\":[\"4\",\"8\",\"12\",\"24\"],\"answer\":1,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"A student photographer receives 175 AED for setting up an event and 28 AED for each photo package sold. An additional bonus equal to 5% of package revenue is paid. What is the least number of packages needed to earn at least 2,500 AED in total?\",\"options\":[\"78\",\"79\",\"80\",\"85\"],\"answer\":2,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"domain\":\"Algebra\",\"prompt\":\"Which value of x satisfies 3(x \\u2212 2) = 15?\",\"options\":[\"3\",\"5\",\"7\",\"13\"],\"answer\":2,\"drill\":\"Common Mistakes\",\"hint\":\"Test each choice in the original equation, including the parentheses.\",\"explanation\":\"For x = 7, 3(7 \\u2212 2) = 3(5) = 15. A common mistake is to ignore the subtraction inside the parentheses.\"},{\"domain\":\"Algebra\",\"prompt\":\"Adult tickets cost $12 and student tickets cost $7. A group bought 8 tickets for $76. How many student tickets were bought?\",\"options\":[\"2\",\"4\",\"5\",\"6\"],\"answer\":1,\"drill\":\"Challenge Question\",\"hint\":\"Test the answer choices as the number of student tickets.\",\"explanation\":\"With 4 student tickets, the remaining 4 are adult tickets. The total is 4($7) + 4($12) = $28 + $48 = $76.\"}]},{\"title\":\"Functions\",\"questions\":[{\"prompt\":\"The functions f and g are defined by f(x) = 2x \\u2212 3 and g(x) = 1\\\/x. Calculate 2f(3) + 3g(3).\",\"options\":[\"10\\\/3\",\"4\",\"7\",\"29\\\/3\"],\"answer\":2,\"drill\":\"Drill 1\",\"hint\":\"Evaluate f(3) and g(3) separately before applying the outside coefficients.\",\"explanation\":\"f(3) = 2(3) \\u2212 3 = 3 and g(3) = 1\\\/3. Therefore 2f(3) + 3g(3) = 2(3) + 3(1\\\/3) = 6 + 1 = 7, choice C.\"},{\"prompt\":\"For every x \\u003E 0, the function g assigns 85% of x to its output; that is, g(x) = 0.85x. Which statement best classifies g?\",\"options\":[\"Decreasing exponential\",\"Decreasing linear\",\"Increasing exponential\",\"Increasing linear\"],\"answer\":3,\"drill\":\"Drill 1\",\"hint\":\"Translate \\u201c85% of x\\u201d into an algebraic expression.\",\"explanation\":\"The definition gives g(x) = 0.85x. This is linear with positive slope 0.85, so it is an increasing linear function, choice D.\"},{\"prompt\":\"The profit P(x), in thousands of dollars, from selling x hundred items is modeled by a quadratic function. The maximum profit is $72,000 when 6 hundred items are sold. Which equation could represent P(x)?\",\"options\":[\"P(x) = 2(x \\u2212 6)\\u00b2 + 72\",\"P(x) = \\u22122(x \\u2212 6)\\u00b2 + 72\",\"P(x) = \\u22122(x + 6)\\u00b2 + 72\",\"P(x) = 2(x + 6)\\u00b2 \\u2212 72\"],\"answer\":1,\"drill\":\"Drill 1\",\"hint\":\"Use vertex form P(x) = a(x \\u2212 h)\\u00b2 + k. A maximum requires a negative value of a.\",\"explanation\":\"The vertex must be (6, 72), so the equation has the form P(x) = a(x \\u2212 6)\\u00b2 + 72. Because 72 is a maximum, the parabola opens downward and a is negative. Choice B fits both conditions.\"},{\"prompt\":\"If g(1) = 24 and g decreases 25% per step, what is g(3)?\",\"options\":[\"12\",\"13.5\",\"18\",\"21\"],\"answer\":1,\"drill\":\"Drill 1\",\"hint\":\"A 25% decrease means multiplying by 0.75 for each increase of 1 in x.\",\"explanation\":\"From x = 1 to x = 3 there are two decreases. Thus g(3) = 24(0.75)\\u00b2 = 24(0.5625) = 13.5.\"},{\"prompt\":\"A function has the values f(0) = 200, f(2) = 140, f(4) = 98, and f(6) = 68.6. Each time x increases by 2, the output decreases by p%. Determine p.\",\"options\":[\"30\",\"43\",\"60\",\"70\"],\"answer\":0,\"drill\":\"Drill 1\",\"hint\":\"Compare one output to the preceding output when x increases by 2.\",\"explanation\":\"When x rises from 0 to 2, f(x) changes from 200 to 140. The decrease is 60, and 60\\\/200 = 0.30 = 30%. The same ratio continues in the table, so p = 30, choice A.\"},{\"prompt\":\"Let g(x) = 2x \\u2212 3. Determine the output of the function when x = 1.\",\"options\":[\"\\u22123\",\"\\u22121\",\"1\",\"5\"],\"answer\":1,\"drill\":\"Drill 2\",\"hint\":\"Substitute x = 1 into g(x) = 2x \\u2212 3.\",\"explanation\":\"g(1) = 2(1) \\u2212 3 = 2 \\u2212 3 = \\u22121.\"},{\"prompt\":\"The function h is defined by h(x) = 12x. Find the input x for which the output is 96.\",\"options\":[\"6\",\"8\",\"10\",\"12\"],\"answer\":1,\"drill\":\"Drill 2\",\"hint\":\"Set 12x equal to 96 and divide.\",\"explanation\":\"12x = 96, so x = 96\\\/12 = 8.\"},{\"prompt\":\"For the function g(x) = x\\u2074, calculate the value of g(3).\",\"options\":[\"7\",\"12\",\"64\",\"81\"],\"answer\":3,\"drill\":\"Drill 2\",\"hint\":\"Raise 3 to the fourth power.\",\"explanation\":\"g(3) = 3\\u2074 = 3\\u00b73\\u00b73\\u00b73 = 81, choice D.\"},{\"prompt\":\"A linear function g has the values g(1) = 2, g(2) = 6, g(3) = 10, and g(4) = 14. Which equation represents g?\",\"options\":[\"g(x)=2x\\u22121\",\"g(x)=3x\\u22121\",\"g(x)=4x\\u22122\",\"g(x)=5x\\u22123\"],\"answer\":2,\"drill\":\"Drill 2\",\"hint\":\"Find the constant change in g(x) for each increase of 1 in x.\",\"explanation\":\"The outputs rise by 4 each time, so the slope is 4. Using g(1) = 2 gives 2 = 4(1) + b, so b = \\u22122. Thus g(x) = 4x \\u2212 2, choice C.\"},{\"prompt\":\"The function f is defined by f(x) = 56\\\/x. For what value of x does f(x) = 7?\",\"options\":[\"\\u22128\",\"7\",\"8\",\"49\"],\"answer\":2,\"drill\":\"Drill 2\",\"hint\":\"Set 56\\\/x equal to 7, then multiply both sides by x.\",\"explanation\":\"56\\\/x = 7 gives 56 = 7x. Dividing by 7 gives x = 8, choice C.\"},{\"prompt\":\"The quadratic function g is defined by g(x) = 2x\\u00b2 \\u2212 2. If g(a) = 70, which listed value could be a?\",\"options\":[\"\\u22128\",\"\\u22126\",\"18\",\"35\"],\"answer\":1,\"drill\":\"Drill 2\",\"hint\":\"Solve 2a\\u00b2 \\u2212 2 = 70.\",\"explanation\":\"2a\\u00b2 \\u2212 2 = 70 gives 2a\\u00b2 = 72 and a\\u00b2 = 36, so a = \\u00b16. The listed possible value is \\u22126, choice B.\"},{\"prompt\":\"The function g is defined by g(x) = \\u221a(x + 4)\\\/3. For which value of x is g(x) = 2?\",\"options\":[\"8\",\"18\",\"32\",\"40\"],\"answer\":2,\"drill\":\"Drill 2\",\"hint\":\"Set \\u221a(x + 4)\\\/3 equal to 2, isolate the square root, then square both sides.\",\"explanation\":\"\\u221a(x + 4)\\\/3 = 2 gives \\u221a(x + 4) = 6. Squaring gives x + 4 = 36, so x = 32, choice C.\"},{\"prompt\":\"The function h is defined by h(x) = (x \\u2212 1)\\\/(x\\u00b2 + 2x \\u2212 5). What is h(3)?\",\"options\":[\"\\u22121\\\/5\",\"1\\\/5\",\"1\\\/2\",\"2\"],\"answer\":1,\"drill\":\"Drill 2\",\"hint\":\"Substitute 3 into the numerator and denominator before simplifying.\",\"explanation\":\"The numerator is 3 \\u2212 1 = 2. The denominator is 3\\u00b2 + 2(3) \\u2212 5 = 9 + 6 \\u2212 5 = 10. Therefore h(3) = 2\\\/10 = 1\\\/5, choice B.\"},{\"prompt\":\"A linear function h passes through the points (\\u22122, 1) and (0, \\u22125). Which equation defines h?\",\"options\":[\"h(x)=\\u22123x\\u22125\",\"h(x)=\\u22123x+1\",\"h(x)=\\u22122x\\u22125\",\"h(x)=\\u2212x\\u22125\"],\"answer\":0,\"drill\":\"Drill 2\",\"hint\":\"Use h(0) for the intercept and the two points to find slope.\",\"explanation\":\"Because h(0) = \\u22125, the y-intercept is \\u22125. The slope from (\\u22122, 1) to (0, \\u22125) is (\\u22125\\u22121)\\\/(0\\u2212(\\u22122)) = \\u22126\\\/2 = \\u22123. Thus h(x) = \\u22123x \\u2212 5, choice A.\"},{\"prompt\":\"A quadratic function g has the values g(\\u22121) = 1, g(0) = \\u22124, and g(2) = \\u22122. Which equation defines g?\",\"options\":[\"g(x) = x\\u00b2 \\u2212 3x \\u2212 4\",\"g(x) = 2x\\u00b2 + 3x \\u2212 4\",\"g(x) = 2x\\u00b2 \\u2212 3x \\u2212 4\",\"g(x) = \\u22122x\\u00b2 \\u2212 3x + 4\"],\"answer\":2,\"drill\":\"Drill 2\",\"hint\":\"Use g(0) to check the constant term, then test x = \\u22121 or x = 2.\",\"explanation\":\"Choice C gives g(0) = \\u22124, g(\\u22121) = 2(1) + 3 \\u2212 4 = 1, and g(2) = 2(4) \\u2212 6 \\u2212 4 = \\u22122. It matches all three values, so g(x) = 2x\\u00b2 \\u2212 3x \\u2212 4.\"},{\"prompt\":\"The functions f and g are defined by f(x) = 2x \\u2212 3 and g(x) = 1\\\/x. Calculate 2f(3) + 3g(3).\",\"options\":[\"10\\\/3\",\"4\",\"7\",\"29\\\/3\"],\"answer\":2,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"A function has the values f(0) = 200, f(2) = 140, f(4) = 98, and f(6) = 68.6. Each time x increases by 2, the output decreases by p%. Determine p.\",\"options\":[\"30\",\"43\",\"60\",\"70\"],\"answer\":0,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"For the function g(x) = x\\u2074, calculate the value of g(3).\",\"options\":[\"7\",\"12\",\"64\",\"81\"],\"answer\":3,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"The function g is defined by g(x) = \\u221a(x + 4)\\\/3. For which value of x is g(x) = 2?\",\"options\":[\"8\",\"18\",\"32\",\"40\"],\"answer\":2,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"A quadratic function g has the values g(\\u22121) = 1, g(0) = \\u22124, and g(2) = \\u22122. Which equation defines g?\",\"options\":[\"g(x) = x\\u00b2 \\u2212 3x \\u2212 4\",\"g(x) = 2x\\u00b2 + 3x \\u2212 4\",\"g(x) = 2x\\u00b2 \\u2212 3x \\u2212 4\",\"g(x) = \\u22122x\\u00b2 \\u2212 3x + 4\"],\"answer\":2,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"domain\":\"Advanced Math\",\"prompt\":\"If f(x) = 2x\\u00b2 \\u2212 1, what is f(3)?\",\"options\":[\"11\",\"17\",\"35\",\"36\"],\"answer\":1,\"drill\":\"Common Mistakes\",\"hint\":\"Replace every x with 3 before applying the operations.\",\"explanation\":\"f(3) = 2(3\\u00b2) \\u2212 1 = 18 \\u2212 1 = 17. A common mistake is to square 2 \\u00d7 3 instead of squaring only the input.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"Let f(x) = 2x \\u2212 3 and g(x) = x\\u00b2 + 1. What is f(g(\\u22122))?\",\"options\":[\"3\",\"5\",\"7\",\"13\"],\"answer\":2,\"drill\":\"Challenge Question\",\"hint\":\"Evaluate the inside function g first.\",\"explanation\":\"g(\\u22122) = (\\u22122)\\u00b2 + 1 = 5. Then f(5) = 2(5) \\u2212 3 = 7.\"}]},{\"title\":\"Linear Equations\",\"questions\":[{\"prompt\":\"The variables x and y satisfy the equation 7(x \\u2212 y) = 2. What is the value of x \\u2212 y?\",\"options\":[\"2\\\/7\",\"2\",\"7\\\/2\",\"7\"],\"answer\":0,\"drill\":\"Drill 1\",\"hint\":\"Divide both sides of 7(x \\u2212 y) = 2 by 7.\",\"explanation\":\"Dividing by 7 isolates the requested expression: x \\u2212 y = 2\\\/7, choice A.\"},{\"prompt\":\"The equation |3x + 3| = 126 has two possible values for x + 1. What is the negative value of x + 1?\",\"options\":[\"\\u221242\",\"\\u221241\",\"41\",\"42\"],\"answer\":0,\"drill\":\"Drill 1\",\"hint\":\"Factor 3 inside the absolute value first.\",\"explanation\":\"|3x + 3| = |3(x + 1)| = 3|x + 1| = 126. Thus |x + 1| = 42, and the requested negative value is \\u221242.\"},{\"prompt\":\"The equation x\\\/y + 3 = 7 \\u2212 2\\\/y relates x and y, where y \\u2260 0. Which expression gives x in terms of y?\",\"options\":[\"x = 4y \\u2212 2\",\"x = 4y + 2\",\"x = 10y \\u2212 2\",\"x = 4 \\u2212 2y\"],\"answer\":0,\"drill\":\"Drill 1\",\"hint\":\"Subtract 3 from both sides, then multiply the entire equation by y.\",\"explanation\":\"Subtracting 3 gives x\\\/y = 4 \\u2212 2\\\/y. Multiplying every term by y gives x = 4y \\u2212 2, choice A.\"},{\"prompt\":\"The point (39, y) satisfies both inequalities x + 8y \\u003E 24 and x \\u003E 30. Which of the following could be the value of y?\",\"options\":[\"\\u22127\",\"\\u22124\",\"\\u22122\",\"\\u22121\"],\"answer\":3,\"drill\":\"Drill 1\",\"hint\":\"Substitute x = 39 into the first inequality.\",\"explanation\":\"39 + 8y \\u003E 24 gives 8y \\u003E \\u221215, so y \\u003E \\u221215\\\/8 = \\u22121.875. Among the choices, only \\u22121 satisfies this. Also 39 \\u003E 30 is true, so choice D works.\"},{\"prompt\":\"A supplier models two monthly charges by A(x) = (13\\\/4)x + 22 and B(x) = kx + 31, where x is the number of units ordered. The charges are never equal for any real value of x. What is k?\",\"options\":[\"\\u221213\\\/4\",\"13\\\/4\",\"9\",\"22\\\/31\"],\"answer\":1,\"drill\":\"Drill 1\",\"hint\":\"Two distinct parallel lines never intersect.\",\"explanation\":\"For the charges never to agree, their slopes must be equal and their intercepts different. The intercepts are already 22 and 31, so k = 13\\\/4.\"},{\"prompt\":\"What is the solution to x \\u2212 16 = 20?\",\"options\":[\"4\",\"16\",\"20\",\"36\"],\"answer\":3,\"drill\":\"Drill 2\",\"hint\":\"Add 16 to both sides.\",\"explanation\":\"x \\u2212 16 = 20, so x = 20 + 16 = 36.\"},{\"prompt\":\"A number n satisfies 3n \\u2212 9 = 21. Determine the value of n \\u2212 3.\",\"options\":[\"3\",\"7\",\"10\",\"12\"],\"answer\":1,\"drill\":\"Drill 2\",\"hint\":\"Solve for n first, then subtract 3.\",\"explanation\":\"3n \\u2212 9 = 21 gives 3n = 30 and n = 10. Therefore n \\u2212 3 = 7.\"},{\"prompt\":\"The equation 8x \\u2212 2 = 12 is true. Based on this equation, find the value of 4x \\u2212 4.\",\"options\":[\"\\u22126\",\"3\",\"6\",\"14\"],\"answer\":1,\"drill\":\"Drill 2\",\"hint\":\"Solve 8x \\u2212 2 = 12, or notice that 4x is half of 8x.\",\"explanation\":\"8x \\u2212 2 = 12 gives 8x = 14, so 4x = 7. Therefore 4x \\u2212 4 = 3, choice B.\"},{\"prompt\":\"Positive numbers a, c, and z satisfy 17a\\\/z = 1\\\/(3c). Which equation writes c in terms of a and z?\",\"options\":[\"c=51az\",\"c=z\\u221251a\",\"c=3az\\\/17\",\"c=z\\\/(51a)\"],\"answer\":3,\"drill\":\"Drill 2\",\"hint\":\"Cross-multiply, then divide by the coefficient of c.\",\"explanation\":\"From 17a\\\/z = 1\\\/(3c), cross-multiplication gives 51ac = z. Dividing by 51a yields c = z\\\/(51a), choice D.\"},{\"prompt\":\"Suppose k satisfies 15k \\u2212 5 = 3(7 \\u2212 k) + 5k. Calculate 13k.\",\"options\":[\"13\",\"21\",\"26\",\"39\"],\"answer\":2,\"drill\":\"Drill 2\",\"hint\":\"Expand 3(7 \\u2212 k), then collect the k terms.\",\"explanation\":\"15k \\u2212 5 = 21 \\u2212 3k + 5k = 21 + 2k. Subtracting 2k and adding 5 gives 13k = 26.\"},{\"prompt\":\"The numbers g and h satisfy 11 = 3g \\u2212 8\\\/h, where g \\u2260 0 and h \\u003C 11. Which equation isolates h in terms of g?\",\"options\":[\"h=8\\\/(3g\\u221211)\",\"h=3g\\\/8\\u221211\",\"h=3g\\\/8+11\",\"h=(3g\\u221211)\\\/8\"],\"answer\":0,\"drill\":\"Drill 2\",\"hint\":\"Move 11, then isolate the reciprocal term 8\\\/h.\",\"explanation\":\"From 11 = 3g \\u2212 8\\\/h, we get 8\\\/h = 3g \\u2212 11. Multiply by h and divide by 3g \\u2212 11 to obtain h = 8\\\/(3g \\u2212 11), choice A.\"},{\"prompt\":\"A number z satisfies 2\\\/(z + 1) = 1\\\/(z \\u2212 2). What is the value of 2\\\/z?\",\"options\":[\"5\",\"5\\\/2\",\"2\",\"2\\\/5\"],\"answer\":3,\"drill\":\"Drill 2\",\"hint\":\"Cross-multiply to solve for z.\",\"explanation\":\"From 2\\\/(z + 1) = 1\\\/(z \\u2212 2), cross-multiplication gives 2(z \\u2212 2) = z + 1. Thus 2z \\u2212 4 = z + 1 and z = 5. Therefore 2\\\/z = 2\\\/5, choice D.\"},{\"prompt\":\"The absolute-value equation |31 \\u2212 x| = 7 has two solutions. Which listed number is one possible value of x?\",\"options\":[\"7\",\"24\",\"31\",\"40\"],\"answer\":1,\"drill\":\"Drill 3\",\"hint\":\"Split |31 \\u2212 x| = 7 into two linear equations.\",\"explanation\":\"Either 31 \\u2212 x = 7 or 31 \\u2212 x = \\u22127. These give x = 24 or x = 38. One possible solution is 24.\"},{\"prompt\":\"A positive number a satisfies |a + 3| = 35. Determine the value of 2a + 6.\",\"options\":[\"64\",\"67\",\"70\",\"76\"],\"answer\":2,\"drill\":\"Drill 3\",\"hint\":\"Use a \\u003E 0 to choose the correct absolute-value case.\",\"explanation\":\"Since a \\u003E 0, a + 3 is positive. Thus a + 3 = 35 and a = 32. Therefore 2a + 6 = 64 + 6 = 70.\"},{\"prompt\":\"A number x satisfies the inequality 3x \\u2212 6 \\u2265 18. Which listed value could be x?\",\"options\":[\"3\",\"4\",\"6\",\"9\"],\"answer\":3,\"drill\":\"Drill 3\",\"hint\":\"Isolate x in the inequality.\",\"explanation\":\"3x \\u2212 6 \\u2265 18 gives 3x \\u2265 24, so x \\u2265 8. Of the choices, only 9 satisfies the inequality, choice D.\"},{\"prompt\":\"The equation 7|x \\u2212 12| \\u2212 3|x \\u2212 12| = 64 has two solutions. Find its negative solution.\",\"options\":[\"\\u221216\",\"\\u22124\",\"4\",\"28\"],\"answer\":1,\"drill\":\"Drill 3\",\"hint\":\"Combine the like absolute-value terms first.\",\"explanation\":\"7|x \\u2212 12| \\u2212 3|x \\u2212 12| = 64 becomes 4|x \\u2212 12| = 64, so |x \\u2212 12| = 16. Thus x = 28 or x = \\u22124. The negative solution is \\u22124.\"},{\"prompt\":\"Given that 2m + 3 \\u2264 5, determine the greatest possible value of 2m \\u2212 3.\",\"options\":[\"\\u22123\",\"\\u22121\",\"1\",\"2\"],\"answer\":1,\"drill\":\"Drill 3\",\"hint\":\"Transform the given inequality directly into one involving 2m \\u2212 3.\",\"explanation\":\"From 2m + 3 \\u2264 5, subtract 6 from both sides to get 2m \\u2212 3 \\u2264 \\u22121. Therefore the greatest possible value is \\u22121, choice B.\"},{\"prompt\":\"How many solutions does y + 12 = y \\u2212 12 have?\",\"options\":[\"Zero\",\"Exactly one\",\"Exactly two\",\"Infinitely many\"],\"answer\":0,\"drill\":\"Drill 4\",\"hint\":\"Subtract y from both sides and check whether the remaining statement is true.\",\"explanation\":\"Subtracting y from both sides gives 12 = \\u221212, which is false. Therefore no value of y satisfies the equation, so it has zero solutions, choice A.\"},{\"prompt\":\"The ordered pair (x, y) satisfies both equations \\u22122x + y = 33 and 3x \\u2212 2y = 16. Find x \\u2212 y.\",\"options\":[\"\\u221249\",\"\\u221233\",\"49\",\"82\"],\"answer\":2,\"drill\":\"Drill 4\",\"hint\":\"Solve the first equation for y, then substitute into the second equation.\",\"explanation\":\"From \\u22122x + y = 33, y = 33 + 2x. Substitute into 3x \\u2212 2y = 16: 3x \\u2212 2(33 + 2x) = 16, so \\u2212x \\u2212 66 = 16 and x = \\u221282. Then y = 33 + 2(\\u221282) = \\u2212131. Thus x \\u2212 y = \\u221282 \\u2212 (\\u2212131) = 49.\"},{\"prompt\":\"The ordered pair (x, y) is the solution to the system 19x + 4y = 41 and 7x + 4y = 17. Determine x + y.\",\"options\":[\"\\u22125\\\/4\",\"1\\\/4\",\"5\\\/4\",\"11\\\/4\"],\"answer\":3,\"drill\":\"Drill 4\",\"hint\":\"Subtract the second equation from the first to eliminate y.\",\"explanation\":\"Subtracting 7x + 4y = 17 from 19x + 4y = 41 gives 12x = 24, so x = 2. Substitution gives 14 + 4y = 17, so y = 3\\\/4. Therefore x + y = 2 + 3\\\/4 = 11\\\/4, choice D.\"},{\"prompt\":\"The equation 5x \\u2212 c = (15x \\u2212 12)\\\/3 is true for infinitely many values of x, where c is a constant. Find c.\",\"options\":[\"3\",\"4\",\"5\",\"12\"],\"answer\":1,\"drill\":\"Drill 4\",\"hint\":\"Simplify the right side and match its constant term to the left side.\",\"explanation\":\"The right side simplifies to (15x \\u2212 12)\\\/3 = 5x \\u2212 4. For 5x \\u2212 c = 5x \\u2212 4 to be true for every x, the constants must match, so c = 4, choice B.\"},{\"prompt\":\"The equation 4x \\u2212 3x(c \\u2212 1) = 2x has infinitely many solutions, where c is a constant. Determine c.\",\"options\":[\"\\u22121\\\/3\",\"2\\\/3\",\"1\",\"5\\\/3\"],\"answer\":3,\"drill\":\"Drill 4\",\"hint\":\"Expand the left side and make its coefficient of x equal to 2.\",\"explanation\":\"Expanding gives 4x \\u2212 3x(c \\u2212 1) = (4 \\u2212 3c + 3)x = (7 \\u2212 3c)x. For infinitely many solutions, 7 \\u2212 3c = 2. Thus \\u22123c = \\u22125 and c = 5\\\/3, choice D.\"},{\"prompt\":\"A quiz has 6 multiple-choice questions and 4 short-response questions worth 3 more points each than a multiple-choice question. The quiz is worth 52 points total. How many points is each multiple-choice question worth?\",\"options\":[\"3\",\"4\",\"5\",\"7\"],\"answer\":1,\"drill\":\"Drill 4\",\"hint\":\"Let m be the value of one multiple-choice question. A short-response question is worth m + 3.\",\"explanation\":\"The total is 6m + 4(m + 3) = 52. Thus 10m + 12 = 52, so 10m = 40 and m = 4 points, choice B.\"},{\"prompt\":\"If 5x + 7 = c(x + d) has infinitely many solutions, what is d?\",\"options\":[\"5\\\/7\",\"1\",\"7\\\/5\",\"7\"],\"answer\":2,\"drill\":\"Drill 4\",\"hint\":\"Expand the right side, then match both the x-coefficient and constant term.\",\"explanation\":\"Expanding gives cx + cd. For this to equal 5x + 7 for every x, c = 5 and cd = 7. Therefore 5d = 7, so d = 7\\\/5.\"},{\"prompt\":\"What is the negative solution to 8|x \\u2212 3| \\u2212 11|x \\u2212 3| = \\u221248?\",\"options\":[\"\\u221219\",\"\\u221213\",\"13\",\"19\"],\"answer\":1,\"drill\":\"Drill 5\",\"hint\":\"Combine the two absolute-value terms before solving the two cases.\",\"explanation\":\"Combining like terms gives \\u22123|x \\u2212 3| = \\u221248, so |x \\u2212 3| = 16. Therefore x \\u2212 3 = 16 or x \\u2212 3 = \\u221216, giving x = 19 or x = \\u221213. The negative solution is \\u221213.\"},{\"prompt\":\"For x \\u2260 \\u22124, which expression is equivalent to (15x\\u00b2 + 72x + 48)\\\/(3x + 12)?\",\"options\":[\"5x + 4\",\"5x + 12\",\"15x + 4\",\"5x\\u00b2 + 4\"],\"answer\":0,\"drill\":\"Drill 5\",\"hint\":\"Factor the numerator and denominator, then cancel their common factor.\",\"explanation\":\"The numerator factors as 3(5x + 4)(x + 4), and the denominator is 3(x + 4). Since x \\u2260 \\u22124, the common factors cancel, leaving 5x + 4, choice A.\"},{\"prompt\":\"An amusement park offers half-day passes for $40 each and all-day passes for $80 each. On one day, the park sold 70 passes and collected $4,600. How many of the passes sold were all-day passes?\",\"options\":[\"25\",\"35\",\"45\",\"55\"],\"answer\":2,\"drill\":\"Drill 5\",\"hint\":\"Let h and a be the numbers of half-day and all-day passes, then write one equation for the count and one for the revenue.\",\"explanation\":\"The equations are h + a = 70 and 40h + 80a = 4,600. Dividing the second equation by 40 gives h + 2a = 115. Subtracting h + a = 70 yields a = 45, so the park sold 45 all-day passes.\"},{\"prompt\":\"A cargo ship is carrying a load equal to two-thirds of its maximum weight capacity. After 7 more tons are added, the load equals 75% of the ship\\u2019s maximum capacity. What is the ship\\u2019s maximum cargo capacity, in tons?\",\"options\":[\"48\",\"60\",\"84\",\"96\"],\"answer\":2,\"drill\":\"Drill 5\",\"hint\":\"Let C be the maximum capacity and translate two-thirds and 75% into fractions.\",\"explanation\":\"The ship initially holds (2\\\/3)C tons. After adding 7 tons it holds 75%, or (3\\\/4)C, so (2\\\/3)C + 7 = (3\\\/4)C. The difference between 3\\\/4 and 2\\\/3 is 1\\\/12, giving 7 = C\\\/12. Thus C = 84 tons, choice C.\"},{\"prompt\":\"Which set of three (x, y) values satisfies y \\u003C 6x + 7?\",\"options\":[\"(\\u22122, \\u22126), (1, 13), (4, 31)\",\"(\\u22122, \\u22126), (1, 10), (4, 28)\",\"(\\u22122, \\u22125), (1, 10), (4, 31)\",\"(\\u22122, \\u22124), (1, 10), (4, 28)\"],\"answer\":1,\"drill\":\"Drill 5\",\"hint\":\"For x = \\u22122, 1, and 4, compare each y-value with \\u22125, 13, and 31 respectively.\",\"explanation\":\"For x = \\u22122, 1, and 4, the upper bounds are \\u22125, 13, and 31. Choice B uses y-values \\u22126, 10, and 28, and each is strictly below its corresponding bound. The other choices contain an equal or greater value, so choice B is correct.\"},{\"prompt\":\"The variables x and y satisfy the equation 7(x \\u2212 y) = 2. What is the value of x \\u2212 y?\",\"options\":[\"2\\\/7\",\"2\",\"7\\\/2\",\"7\"],\"answer\":0,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"The equation 8x \\u2212 2 = 12 is true. Based on this equation, find the value of 4x \\u2212 4.\",\"options\":[\"\\u22126\",\"3\",\"6\",\"14\"],\"answer\":1,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"A number x satisfies the inequality 3x \\u2212 6 \\u2265 18. Which listed value could be x?\",\"options\":[\"3\",\"4\",\"6\",\"9\"],\"answer\":3,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"The equation 4x \\u2212 3x(c \\u2212 1) = 2x has infinitely many solutions, where c is a constant. Determine c.\",\"options\":[\"\\u22121\\\/3\",\"2\\\/3\",\"1\",\"5\\\/3\"],\"answer\":3,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"Which set of three (x, y) values satisfies y \\u003C 6x + 7?\",\"options\":[\"(\\u22122, \\u22126), (1, 13), (4, 31)\",\"(\\u22122, \\u22126), (1, 10), (4, 28)\",\"(\\u22122, \\u22125), (1, 10), (4, 31)\",\"(\\u22122, \\u22124), (1, 10), (4, 28)\"],\"answer\":1,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"domain\":\"Algebra\",\"prompt\":\"Which inequality is equivalent to \\u22123x \\u003E 12?\",\"options\":[\"x \\u003E \\u22124\",\"x \\u003C \\u22124\",\"x \\u003E 4\",\"x \\u003C 4\"],\"answer\":1,\"drill\":\"Common Mistakes\",\"hint\":\"Dividing an inequality by a negative number reverses its direction.\",\"explanation\":\"Dividing both sides by \\u22123 reverses \\u003E to \\u003C, giving x \\u003C \\u22124.\"},{\"domain\":\"Algebra\",\"prompt\":\"A school sold 180 tickets. Student tickets cost $6 and adult tickets cost $10. The total revenue was $1,320. How many adult tickets were sold?\",\"options\":[\"45\",\"60\",\"75\",\"120\"],\"answer\":1,\"drill\":\"Challenge Question\",\"hint\":\"Let s + a = 180 and 6s + 10a = 1,320.\",\"explanation\":\"Substitute s = 180 \\u2212 a: 6(180 \\u2212 a) + 10a = 1,320. Then 1,080 + 4a = 1,320, so a = 60.\"}]},{\"title\":\"Nonlinear Equations\",\"questions\":[{\"prompt\":\"Which expression is equivalent to x\\u00b2 \\u2212 13x + 40?\",\"options\":[\"(x\\u22124)(x\\u221210)\",\"(x\\u22125)(x\\u22128)\",\"(x+5)(x+8)\",\"(x\\u22122)(x\\u221220)\"],\"answer\":1,\"drill\":\"Drill 1\",\"hint\":\"Find two numbers whose product is 40 and whose sum is \\u221213.\",\"explanation\":\"The numbers are \\u22125 and \\u22128 because (\\u22125)(\\u22128) = 40 and \\u22125 + (\\u22128) = \\u221213. Therefore x\\u00b2 \\u2212 13x + 40 = (x \\u2212 5)(x \\u2212 8), choice B.\"},{\"prompt\":\"Which expression is equivalent to 7a\\u00b3 \\u2212 3a\\u00b3 + 5a\\u00b2?\",\"options\":[\"4a\\u00b3 + 5a\\u00b2\",\"4a\\u2076 + 5a\\u00b2\",\"9a\\u00b3\",\"4a\\u2075 + 5a\\u00b2\"],\"answer\":0,\"drill\":\"Drill 1\",\"hint\":\"Combine only the like terms with the same variable and exponent.\",\"explanation\":\"The terms 7a\\u00b3 and \\u22123a\\u00b3 are like terms, so they combine to 4a\\u00b3. The 5a\\u00b2 term has a different exponent and remains unchanged. The result is 4a\\u00b3 + 5a\\u00b2, choice A.\"},{\"prompt\":\"The variables a, b, and c are all positive. Which expression is equivalent to (a\\u207b\\u00b2b\\u00b2c\\u2076)(a\\u207b\\u00b3b\\u207b\\u00b9c\\u00b2)?\",\"options\":[\"a\\u207b\\u2075bc\\u2078\",\"ab\\u00b3c\\u2074\",\"a\\u2076b\\u207b\\u00b2c\\u00b9\\u00b2\",\"a\\u00b9\\u00b9b\\u2076c\\u00b3\"],\"answer\":0,\"drill\":\"Drill 1\",\"hint\":\"When multiplying powers with the same base, add their exponents.\",\"explanation\":\"Add exponents base by base: a: \\u22122 + (\\u22123) = \\u22125; b: 2 + (\\u22121) = 1; c: 6 + 2 = 8. Thus the product is a\\u207b\\u2075bc\\u2078, choice A.\"},{\"prompt\":\"The graph of the quadratic function y = f(x) is shown. On which interval is f(x) negative?\",\"options\":[\"x \\u003C \\u22121\",\"\\u22121 \\u003C x \\u003C 3\",\"x \\u003E 3\",\"x \\u003C \\u22121 or x \\u003E 3\"],\"answer\":1,\"drill\":\"Drill 1\",\"hint\":\"The function is negative where the graph lies below the x-axis.\",\"explanation\":\"The graph crosses the x-axis at x = \\u22121 and x = 3. Between these intercepts, the parabola lies below the x-axis, so f(x) \\u003C 0 for \\u22121 \\u003C x \\u003C 3, choice B. The endpoints are not included because f(x) = 0 there.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/nonlinear-equations\\\/drill-1\\\/generated-parabola-graph.svg\"},{\"prompt\":\"The equation x = 70\\\/(x \\u2212 3) has two solutions. What is its negative solution?\",\"options\":[\"\\u221210\",\"\\u22127\",\"7\",\"10\"],\"answer\":1,\"drill\":\"Drill 1\",\"hint\":\"Multiply by x \\u2212 3 and factor the resulting quadratic.\",\"explanation\":\"x = 70\\\/(x \\u2212 3) gives x(x \\u2212 3) = 70, so x\\u00b2 \\u2212 3x \\u2212 70 = 0. Factoring gives (x \\u2212 10)(x + 7) = 0. The negative solution is \\u22127.\"},{\"prompt\":\"What is the product of all values of x that satisfy 5x\\u00b2 \\u2212 20x \\u2212 10 = 0?\",\"options\":[\"\\u22125\\u221a6\",\"\\u22122\",\"2\",\"5\\u221a6\"],\"answer\":1,\"drill\":\"Drill 2\",\"hint\":\"For ax\\u00b2 + bx + c = 0, the product of the roots is c\\\/a.\",\"explanation\":\"For 5x\\u00b2 \\u2212 20x \\u2212 10 = 0, the product of the roots is c\\\/a = \\u221210\\\/5 = \\u22122, choice B.\"},{\"prompt\":\"Which pair gives all solutions to the equation \\u221a(x\\u00b2 \\u2212 6x \\u2212 66) = 5?\",\"options\":[\"\\u22127 and \\u221213\",\"\\u22127 and 13\",\"7 and \\u221213\",\"7 and 13\"],\"answer\":1,\"drill\":\"Drill 2\",\"hint\":\"Square both sides, move 25 to the left, and factor.\",\"explanation\":\"Squaring gives x\\u00b2 \\u2212 6x \\u2212 66 = 25, so x\\u00b2 \\u2212 6x \\u2212 91 = 0. This factors as (x \\u2212 13)(x + 7) = 0, giving x = 13 or \\u22127. Both work in the original equation, so choice B is correct.\"},{\"prompt\":\"Which expression is a factor of 6x\\u00b2 \\u2212 7x \\u2212 3?\",\"options\":[\"3x + 1\",\"3x \\u2212 1\",\"2x + 3\",\"2x + 1\"],\"answer\":0,\"drill\":\"Drill 2\",\"hint\":\"Split the middle term into two terms whose product is 6(\\u22123) = \\u221218.\",\"explanation\":\"6x\\u00b2 \\u2212 7x \\u2212 3 = 6x\\u00b2 \\u2212 9x + 2x \\u2212 3 = 3x(2x \\u2212 3) + 1(2x \\u2212 3) = (3x + 1)(2x \\u2212 3). Therefore 3x + 1 is a factor, choice A.\"},{\"prompt\":\"A bacteria culture starts with 240 cells and grows by 5% every 30 minutes. Which equation gives the number B of cells after m minutes?\",\"options\":[\"B = 240(1.05)^(30m)\",\"B = 240(0.95)^(m\\\/30)\",\"B = 240(1.05)^(m\\\/30)\",\"B = 240 + 1.05(m\\\/30)\"],\"answer\":2,\"drill\":\"Drill 2\",\"hint\":\"The exponent should count how many 30-minute growth periods have passed.\",\"explanation\":\"After m minutes, the number of 30-minute periods is m\\\/30. Each period multiplies the culture by 1.05, so B = 240(1.05)^(m\\\/30), choice C.\"},{\"prompt\":\"A recycling program models the number of participating households by H = 375(1.08)^(4t), where t is time in years. According to this model, by what percentage does participation increase every 3 months?\",\"options\":[\"2\",\"8\",\"24\",\"32\"],\"answer\":1,\"drill\":\"Drill 2\",\"hint\":\"Determine which time interval is counted by the exponent.\",\"explanation\":\"There are four 3-month periods in each year, so 4t counts those periods. Each period multiplies participation by 1.08, representing an 8% increase.\"},{\"prompt\":\"Which of the following expressions is an equivalent factored form of x\\u00b2 + 5x \\u2212 24?\",\"options\":[\"(x+3)(x\\u22128)\",\"(x+4)(x\\u22126)\",\"(x+6)(x\\u22124)\",\"(x+8)(x\\u22123)\"],\"answer\":3,\"drill\":\"Drill 3\",\"hint\":\"Find two numbers with product \\u221224 and sum 5.\",\"explanation\":\"The numbers are 8 and \\u22123. Therefore x\\u00b2 + 5x \\u2212 24 = (x + 8)(x \\u2212 3), choice D.\"},{\"prompt\":\"The equation x = 36\\\/(x \\u2212 9) has two solutions. Which listed value is one possible solution?\",\"options\":[\"\\u221212\",\"\\u22124\",\"3\",\"12\"],\"answer\":3,\"drill\":\"Drill 3\",\"hint\":\"Multiply by x \\u2212 9, move all terms to one side, and factor.\",\"explanation\":\"x = 36\\\/(x \\u2212 9) gives x\\u00b2 \\u2212 9x \\u2212 36 = 0. Factoring gives (x \\u2212 12)(x + 3) = 0, so the solutions are 12 and \\u22123. One possible solution is 12.\"},{\"prompt\":\"Which expression is a factor of 4x\\u00b2 \\u2212 25x + 6?\",\"options\":[\"4x + 1\",\"4x \\u2212 1\",\"x + 6\",\"x \\u2212 2\"],\"answer\":1,\"drill\":\"Drill 3\",\"hint\":\"Find two numbers whose product is 24 and whose sum is \\u221225.\",\"explanation\":\"Split the middle term: 4x\\u00b2 \\u2212 24x \\u2212 x + 6 = 4x(x \\u2212 6) \\u2212 1(x \\u2212 6) = (4x \\u2212 1)(x \\u2212 6). Therefore 4x \\u2212 1 is a factor, choice B.\"},{\"prompt\":\"If x \\u2212 k is a factor of 2x\\u00b3 + 10x\\u00b2 \\u2212 72x and k is positive, what is k?\",\"options\":[\"4\",\"6\",\"9\",\"12\"],\"answer\":0,\"drill\":\"Drill 3\",\"hint\":\"Factor out 2x, then factor the remaining quadratic.\",\"explanation\":\"2x\\u00b3 + 10x\\u00b2 \\u2212 72x = 2x(x\\u00b2 + 5x \\u2212 36) = 2x(x + 9)(x \\u2212 4). The positive zero associated with the factor x \\u2212 k is 4, so k = 4, choice A.\"},{\"prompt\":\"What is the greatest solution to the equation \\u221a(2x + 86) = \\u221a((x + 3)\\u00b2)?\",\"options\":[\"\\u221211\",\"\\u22127\",\"7\",\"11\"],\"answer\":2,\"drill\":\"Drill 3\",\"hint\":\"Square both sides and factor the resulting quadratic.\",\"explanation\":\"Squaring gives 2x + 86 = (x + 3)\\u00b2, so x\\u00b2 + 4x \\u2212 77 = 0. This factors as (x + 11)(x \\u2212 7) = 0, giving \\u221211 and 7. Both check, and the greatest is 7.\"},{\"prompt\":\"In the equation 42x\\u00b2 \\u2212 (42d + c)x + cd = 0, the constants k, c, and d are all positive. If the product of the equation\\u2019s solutions is kcd, what is the value of k?\",\"options\":[\"1\\\/81\",\"1\\\/42\",\"1\\\/39\",\"3\"],\"answer\":1,\"drill\":\"Drill 3\",\"hint\":\"For ax\\u00b2 + bx + c = 0, the product of the roots is c\\\/a.\",\"explanation\":\"The constant term is cd and the leading coefficient is 42, so the product of the roots is cd\\\/42. Since this equals kcd, k = 1\\\/42, choice B.\"},{\"prompt\":\"How many distinct real solutions does 6x\\u00b2 + x \\u2212 2 = 0 have?\",\"options\":[\"Zero\",\"Exactly one\",\"Exactly two\",\"Infinitely many\"],\"answer\":2,\"drill\":\"Drill 4\",\"hint\":\"Use the discriminant b\\u00b2 \\u2212 4ac to determine the number of real roots.\",\"explanation\":\"Here a = 6, b = 1, and c = \\u22122. The discriminant is 1\\u00b2 \\u2212 4(6)(\\u22122) = 1 + 48 = 49. Since 49 is positive, the equation has exactly two distinct real solutions, choice C.\"},{\"prompt\":\"The equation ax\\u00b2 \\u2212 6x = \\u22126 has no real solutions. Which listed number could be the value of a?\",\"options\":[\"\\u22122\",\"\\u22121\",\"1\",\"2\"],\"answer\":3,\"drill\":\"Drill 4\",\"hint\":\"Write the equation in standard form and require its discriminant to be negative.\",\"explanation\":\"The equation becomes ax\\u00b2 \\u2212 6x + 6 = 0. Its discriminant is 36 \\u2212 24a. For no real solutions, 36 \\u2212 24a \\u003C 0, so a \\u003E 3\\\/2. Of the choices, only a = 2 works, choice D.\"},{\"prompt\":\"The system y = x\\u00b2 + 4x + 14 and y = 2k has exactly one real solution, where k is a constant. What is the value of k?\",\"options\":[\"2\",\"4\",\"5\",\"7\"],\"answer\":2,\"drill\":\"Drill 4\",\"hint\":\"Set the equations equal and make the resulting quadratic\\u0027s discriminant zero.\",\"explanation\":\"Setting the equations equal gives x\\u00b2 + 4x + 14 \\u2212 2k = 0. Exactly one real solution requires a zero discriminant: 4\\u00b2 \\u2212 4(1)(14 \\u2212 2k) = 0. Thus 16 \\u2212 56 + 8k = 0, so 8k = 40 and k = 5.\"},{\"prompt\":\"The graph of the quadratic function y = f(x) is shown. The vertex is (1, 4), and the graph passes through (3, 0). What is the value of f(\\u22122)?\",\"options\":[\"\\u22129\",\"\\u22125\",\"\\u22121\",\"5\"],\"answer\":1,\"drill\":\"Drill 4\",\"hint\":\"Write the function in vertex form, f(x) = a(x \\u2212 1)\\u00b2 + 4, and use the point (3, 0) to find a.\",\"explanation\":\"Using vertex form, f(x) = a(x \\u2212 1)\\u00b2 + 4. Since (3, 0) is on the graph, 0 = a(3 \\u2212 1)\\u00b2 + 4, so 4a = \\u22124 and a = \\u22121. Therefore f(\\u22122) = \\u2212(\\u22122 \\u2212 1)\\u00b2 + 4 = \\u22129 + 4 = \\u22125, choice B.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/nonlinear-equations\\\/drill-4\\\/generated-downward-parabola-graph.svg\"},{\"prompt\":\"A savings account earns 3% interest each year. If the account begins with a $1,000 deposit and receives no additional deposits, approximately how much money will it contain after 12 years?\",\"options\":[\"$1,360\",\"$1,405\",\"$1,426\",\"$1,720\"],\"answer\":2,\"drill\":\"Drill 5\",\"hint\":\"Use the compound-growth model A = P(1 + r)\\u1d57.\",\"explanation\":\"The initial deposit is $1,000 and the annual growth factor is 1 + 0.03 = 1.03. After 12 years, A = 1,000(1.03)\\u00b9\\u00b2 \\u2248 1,425.76, which rounds to $1,426, choice C.\"},{\"prompt\":\"Country A had a population of 120,000 at the end of 1990 and then decreased at a constant annual rate. In 2020, its population was approximately 65,460. If P(t) is the population t years after 1990, which equation best models this situation?\",\"options\":[\"P(t) = 120,000(0.98)\\u1d57\",\"P(t) = 65,460(0.98)\\u1d57\",\"P(t) = 120,000(1.02)\\u1d57\",\"P(t) = 65,460(1.02)\\u1d57\"],\"answer\":0,\"drill\":\"Drill 5\",\"hint\":\"Use the 1990 population as the initial value and choose a multiplier below 1 because the population decreases.\",\"explanation\":\"At t = 0, the population is 120,000, so that must be the initial value. A consistent decrease uses a factor below 1. In fact, 120,000(0.98)\\u00b3\\u2070 \\u2248 65,460 for the 30 years from 1990 to 2020. Therefore P(t) = 120,000(0.98)\\u1d57, choice A.\"},{\"prompt\":\"A new bar of hand soap weighs 100 grams. Its weight decreases by 2.5% each time it is used. Approximately how many grams will the bar weigh after 20 uses?\",\"options\":[\"52.62 grams\",\"57.43 grams\",\"60.27 grams\",\"77.85 grams\"],\"answer\":2,\"drill\":\"Drill 5\",\"hint\":\"A 2.5% decrease means the bar retains 97.5% of its weight after each use.\",\"explanation\":\"The decay multiplier is 1 \\u2212 0.025 = 0.975. After 20 uses, the weight is 100(0.975)\\u00b2\\u2070 \\u2248 60.27 grams, choice C.\"},{\"prompt\":\"Karim buys a car for $30,000, and its value decreases by 15% each year. Which equation models the car\\u2019s value v, in dollars, t years after the purchase?\",\"options\":[\"v = 0.15(1 \\u2212 30,000)\\u1d57\",\"v = (1 \\u2212 30,000)\\u1d57\",\"v = 30,000(1 \\u2212 0.15)\\u1d57\",\"v = 30,000(1 \\u2212 0.85)\\u1d57\"],\"answer\":2,\"drill\":\"Drill 5\",\"hint\":\"Multiply the initial value by the fraction of value retained each year.\",\"explanation\":\"The initial value is $30,000. A 15% annual decrease means the car retains 1 \\u2212 0.15 = 0.85 of its value each year. Thus v = 30,000(1 \\u2212 0.15)\\u1d57 = 30,000(0.85)\\u1d57, choice C.\"},{\"prompt\":\"The system x\\u00b2 + y\\u00b2 = 245 and y = 2x has a solution at (x, y). What is y\\u00b2?\",\"options\":[\"49\",\"98\",\"196\",\"245\"],\"answer\":2,\"drill\":\"Drill 6\",\"hint\":\"Substitute y = 2x into the first equation, then use y\\u00b2 = 4x\\u00b2.\",\"explanation\":\"Substituting y = 2x gives x\\u00b2 + (2x)\\u00b2 = 245, so 5x\\u00b2 = 245 and x\\u00b2 = 49. Since y\\u00b2 = (2x)\\u00b2 = 4x\\u00b2, y\\u00b2 = 4(49) = 196, choice C.\"},{\"prompt\":\"Which expression is equivalent to the 18th root of x\\u00b9\\u00b2 for positive x?\",\"options\":[\"x^(2\\\/3)\",\"x^(3\\\/2)\",\"x\\u2076\",\"x\\u00b3\\u2070\"],\"answer\":0,\"drill\":\"Drill 6\",\"hint\":\"Rewrite the radical as a rational exponent and reduce the fraction.\",\"explanation\":\"The 18th root of x\\u00b9\\u00b2 is (x\\u00b9\\u00b2)^(1\\\/18) = x^(12\\\/18). Reducing 12\\\/18 gives 2\\\/3, so the equivalent expression is x^(2\\\/3), choice A.\"},{\"prompt\":\"An object starts from rest and accelerates at a constant rate a. Its displacement after t seconds is modeled by s(t) = (1\\\/2)a\\u00b7t\\u00b2, where a \\u003E 0 and t \\u003E 0. What is the ratio of its displacement at t seconds to its displacement at 2.5t seconds?\",\"options\":[\"4\\\/25\",\"2\\\/5\",\"5\\\/2\",\"25\\\/4\"],\"answer\":0,\"drill\":\"Drill 6\",\"hint\":\"Evaluate the formula at t and at 2.5t, then cancel the common factors.\",\"explanation\":\"At t seconds, s(t) = (1\\\/2)a\\u00b7t\\u00b2. At 2.5t seconds, s(2.5t) = (1\\\/2)a(2.5t)\\u00b2 = (1\\\/2)a(6.25t\\u00b2). Their ratio is 1\\\/6.25 = 4\\\/25, choice A.\"},{\"prompt\":\"What is the positive solution to \\u221a(9x + 28) = \\u221a((x + 2)\\u00b2)?\",\"options\":[\"\\u22128\",\"\\u22123\",\"3\",\"8\"],\"answer\":3,\"drill\":\"Drill 6\",\"hint\":\"Square both sides, solve the resulting quadratic, and select the positive root.\",\"explanation\":\"Squaring gives 9x + 28 = (x + 2)\\u00b2 = x\\u00b2 + 4x + 4. Rearranging yields x\\u00b2 \\u2212 5x \\u2212 24 = 0, which factors as (x \\u2212 8)(x + 3) = 0. The roots are 8 and \\u22123, so the positive solution is 8, choice D.\"},{\"prompt\":\"Which value of b makes 25x\\u00b2 + bx + 4 = 0 have no real solutions?\",\"options\":[\"\\u221220\",\"12\",\"20\",\"100\"],\"answer\":1,\"drill\":\"Drill 6\",\"hint\":\"Require the discriminant b\\u00b2 \\u2212 4ac to be negative.\",\"explanation\":\"The discriminant is b\\u00b2 \\u2212 4(25)(4) = b\\u00b2 \\u2212 400. No real solutions require b\\u00b2 \\u003C 400, or |b| \\u003C 20. Only 12 satisfies this strict inequality; \\u00b120 give exactly one real solution. Therefore choice B is correct.\"},{\"prompt\":\"If 5x\\u00b2 \\u2212 bx + 18 = (px \\u2212 r)(x \\u2212 s), where p, r, and s are integers, which expression must be an integer?\",\"options\":[\"b\\\/p\",\"b\\\/r\",\"18\\\/p\",\"18\\\/r\"],\"answer\":3,\"drill\":\"Drill 7\",\"hint\":\"Expand the factored form and compare its leading and constant coefficients.\",\"explanation\":\"Expanding gives px\\u00b2 \\u2212 (ps + r)x + rs. Comparing coefficients shows p = 5 and rs = 18. Since r and s are integers and 18\\\/r = rs\\\/r = s, 18\\\/r must be an integer. Therefore choice D is correct.\"},{\"prompt\":\"If x \\u2212 a is a factor of 3x\\u2074 \\u2212 48x\\u00b3 + 189x\\u00b2 and a is positive, what is the greatest possible value of a?\",\"options\":[\"3\",\"7\",\"9\",\"21\"],\"answer\":2,\"drill\":\"Drill 7\",\"hint\":\"Factor out 3x\\u00b2, then factor the remaining quadratic.\",\"explanation\":\"Factor the expression as 3x\\u00b2(x\\u00b2 \\u2212 16x + 63). The quadratic factors as (x \\u2212 7)(x \\u2212 9), so the positive values of a are 7 and 9. The greatest possible value is 9, choice C.\"},{\"prompt\":\"A geometric sequence begins with \\u221212, and each term after the first is one-third of the preceding term. Which expression represents its nth term?\",\"options\":[\"\\u221212(1\\\/3)\\u207f\",\"\\u221212(1\\\/3)\\u207f\\u207b\\u00b9\",\"(1\\\/3)(\\u221212)\\u207f\",\"(1\\\/3)(\\u221212)\\u207f\\u207b\\u00b9\"],\"answer\":1,\"drill\":\"Drill 7\",\"hint\":\"Use a\\u2099 = a\\u2081r\\u207f\\u207b\\u00b9 with first term \\u221212 and common ratio 1\\\/3.\",\"explanation\":\"A geometric sequence with first term a\\u2081 and ratio r has nth term a\\u2099 = a\\u2081r\\u207f\\u207b\\u00b9. Here a\\u2081 = \\u221212 and r = 1\\\/3, so a\\u2099 = \\u221212(1\\\/3)\\u207f\\u207b\\u00b9, choice B.\"},{\"prompt\":\"For positive a, the product of the sixth root of 2\\u00b9\\u00b2a\\u00b9\\u2078 and 2 times the cube root of 4\\u00b3a\\u2076 can be written as ca\\u1d4f. What is c \\u2212 k?\",\"options\":[\"5\",\"16\",\"27\",\"37\"],\"answer\":2,\"drill\":\"Drill 7\",\"hint\":\"Rewrite each radical using rational exponents, simplify the numerical factors, and add the powers of a.\",\"explanation\":\"The sixth root of 2\\u00b9\\u00b2a\\u00b9\\u2078 is 2\\u00b2a\\u00b3 = 4a\\u00b3. Also, 2 times the cube root of 4\\u00b3a\\u2076 is 2\\u00b74a\\u00b2 = 8a\\u00b2. Their product is 32a\\u2075, so c = 32 and k = 5. Thus c \\u2212 k = 27, choice C.\"},{\"prompt\":\"One solution of x\\u00b2 + 6x + 2 = 0 is \\u22123 \\u2212 \\u221ac. What is c?\",\"options\":[\"5\",\"7\",\"9\",\"28\"],\"answer\":1,\"drill\":\"Drill 7\",\"hint\":\"Use the quadratic formula or complete the square.\",\"explanation\":\"The quadratic formula gives x = (\\u22126 \\u00b1 \\u221a(36 \\u2212 8))\\\/2 = (\\u22126 \\u00b1 \\u221a28)\\\/2 = \\u22123 \\u00b1 \\u221a7. Comparing \\u22123 \\u2212 \\u221a7 with \\u22123 \\u2212 \\u221ac shows c = 7, choice B.\"},{\"prompt\":\"Which expression is equivalent to x\\u00b2 \\u2212 13x + 40?\",\"options\":[\"(x\\u22124)(x\\u221210)\",\"(x\\u22125)(x\\u22128)\",\"(x+5)(x+8)\",\"(x\\u22122)(x\\u221220)\"],\"answer\":1,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"A bacteria culture starts with 240 cells and grows by 5% every 30 minutes. Which equation gives the number B of cells after m minutes?\",\"options\":[\"B = 240(1.05)^(30m)\",\"B = 240(0.95)^(m\\\/30)\",\"B = 240(1.05)^(m\\\/30)\",\"B = 240 + 1.05(m\\\/30)\"],\"answer\":2,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"The equation ax\\u00b2 \\u2212 6x = \\u22126 has no real solutions. Which listed number could be the value of a?\",\"options\":[\"\\u22122\",\"\\u22121\",\"1\",\"2\"],\"answer\":3,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"Which expression is equivalent to the 18th root of x\\u00b9\\u00b2 for positive x?\",\"options\":[\"x^(2\\\/3)\",\"x^(3\\\/2)\",\"x\\u2076\",\"x\\u00b3\\u2070\"],\"answer\":0,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"One solution of x\\u00b2 + 6x + 2 = 0 is \\u22123 \\u2212 \\u221ac. What is c?\",\"options\":[\"5\",\"7\",\"9\",\"28\"],\"answer\":1,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"domain\":\"Advanced Math\",\"prompt\":\"If \\u221a(x \\u2212 1) = 5, what is the value of x?\",\"options\":[\"6\",\"24\",\"25\",\"26\"],\"answer\":3,\"drill\":\"Common Mistakes\",\"hint\":\"Square both sides first, then isolate x.\",\"explanation\":\"Squaring gives x \\u2212 1 = 25, so x = 26. A common mistake is to stop at 25 and forget the \\u22121 inside the radical.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"For what value of k does x\\u00b2 \\u2212 (k + 2)x + 2k = 0 have exactly one real solution?\",\"options\":[\"\\u22122\",\"0\",\"2\",\"4\"],\"answer\":2,\"drill\":\"Challenge Question\",\"hint\":\"Exactly one real solution means the discriminant equals zero.\",\"explanation\":\"The discriminant is (k + 2)\\u00b2 \\u2212 8k = k\\u00b2 \\u2212 4k + 4 = (k \\u2212 2)\\u00b2. It equals zero when k = 2.\"}]},{\"title\":\"Percentages \\u0026 Ratios\",\"questions\":[{\"prompt\":\"If 24 is x percent of 15, what is x?\",\"options\":[\"1.6\",\"9\",\"62.5\",\"160\"],\"answer\":3,\"drill\":\"Drill 1\",\"hint\":\"Write 24 = (x\\\/100)(15) and solve for x.\",\"explanation\":\"24 = (x\\\/100)(15). Multiplying by 100 and dividing by 15 gives x = (24\\\/15)(100) = 160. Therefore 24 is 160% of 15, choice D.\"},{\"prompt\":\"Which equation represents: 27 is 5 less than 4 times n?\",\"options\":[\"27 = (\\u22121)n\",\"27 = 4n \\u2212 5\",\"27 \\u2212 5 = 4n\",\"27 = 5n \\u2212 4\"],\"answer\":1,\"drill\":\"Drill 1\",\"hint\":\"Translate \\u201c4 times n\\u201d first, then subtract 5.\",\"explanation\":\"Four times n is 4n. Five less than that is 4n \\u2212 5. Since this quantity equals 27, the equation is 27 = 4n \\u2212 5, choice B.\"},{\"prompt\":\"A student answered 36 of 45 questions correctly. What percentage of the questions did the student answer correctly?\",\"options\":[\"75%\",\"80%\",\"81%\",\"90%\"],\"answer\":1,\"drill\":\"Drill 1\",\"hint\":\"Divide the number correct by the total number of questions, then multiply by 100%.\",\"explanation\":\"The fraction correct is 36\\\/45 = 0.8. Converting to a percentage gives 0.8 \\u00d7 100% = 80%, choice B.\"},{\"prompt\":\"An investment of $18,000 earns 4.5% simple interest in one year. What is the value of the investment after one year?\",\"options\":[\"$810\",\"$18,045\",\"$18,450\",\"$18,810\"],\"answer\":3,\"drill\":\"Drill 1\",\"hint\":\"Calculate 4.5% of $18,000 and add it to the original investment.\",\"explanation\":\"The interest is 0.045($18,000) = $810. The value after one year is $18,000 + $810 = $18,810, choice D.\"},{\"prompt\":\"A fundraiser sells tickets for $32 each and adds a 6.25% service charge to each ticket. What is the minimum number of tickets needed to collect at least $1,500?\",\"options\":[\"43\",\"44\",\"45\",\"47\"],\"answer\":2,\"drill\":\"Drill 1\",\"hint\":\"Find the amount collected per ticket, divide $1,500 by that amount, and round up.\",\"explanation\":\"The service charge is 0.0625($32) = $2, so each ticket collects $34. Since 1,500\\\/34 \\u2248 44.12, at least 45 whole tickets are required, choice C.\"},{\"prompt\":\"After a 20% discount, a jacket costs $72. What was the original price of the jacket?\",\"options\":[\"$86.40\",\"$90\",\"$92\",\"$96\"],\"answer\":1,\"drill\":\"Drill 2\",\"hint\":\"After a 20% discount, the sale price is 80% of the original price.\",\"explanation\":\"Let p be the original price. The sale price is 0.80p, so 0.80p = 72. Dividing by 0.80 gives p = 90. The original price was $90, choice B.\"},{\"prompt\":\"A quantity is increased by 25% and then the new value is decreased by 20%. Compared with the original quantity, what is the final percentage change?\",\"options\":[\"A 5% decrease\",\"No change\",\"A 5% increase\",\"A 10% increase\"],\"answer\":1,\"drill\":\"Drill 2\",\"hint\":\"Apply the two multipliers successively: 1.25 and 0.80.\",\"explanation\":\"The combined multiplier is 1.25 \\u00d7 0.80 = 1.00. Therefore the final value equals the original value, so there is no percentage change, choice B.\"},{\"prompt\":\"In a club, the ratio of juniors to seniors is 3:5. If there are 40 seniors, how many total members are in the club?\",\"options\":[\"24\",\"48\",\"64\",\"72\"],\"answer\":2,\"drill\":\"Drill 2\",\"hint\":\"Five ratio parts represent 40 seniors. Find the value of one part first.\",\"explanation\":\"Since 5 parts equal 40, each part equals 8. The total ratio has 3 + 5 = 8 parts, so the club has 8 \\u00d7 8 = 64 members, choice C.\"},{\"prompt\":\"On a map, 1 inch represents 18 miles. Two towns are 2.75 inches apart on the map. What is their actual distance?\",\"options\":[\"20.75 miles\",\"36 miles\",\"49.5 miles\",\"54 miles\"],\"answer\":2,\"drill\":\"Drill 2\",\"hint\":\"Multiply the map distance by the number of miles represented by each inch.\",\"explanation\":\"The actual distance is 2.75 \\u00d7 18 = 49.5 miles, choice C.\"},{\"prompt\":\"A fruit drink contains 18 liters of juice, which is 30% of the total volume. What is the total volume of the drink?\",\"options\":[\"42 liters\",\"54 liters\",\"60 liters\",\"72 liters\"],\"answer\":2,\"drill\":\"Drill 2\",\"hint\":\"Set 30% of the total volume equal to 18 liters.\",\"explanation\":\"If V is the total volume, then 0.30V = 18. Thus V = 18\\\/0.30 = 60 liters, choice C.\"},{\"prompt\":\"A tablet has an original price of $250. It is discounted by 15%, and then 8% sales tax is applied to the discounted price. What is the final price?\",\"options\":[\"$212.50\",\"$227.00\",\"$229.50\",\"$232.50\"],\"answer\":2,\"drill\":\"Drill 2\",\"hint\":\"Multiply by 0.85 for the discount, then by 1.08 for the tax.\",\"explanation\":\"After the discount, the price is 250(0.85) = $212.50. Applying tax gives 212.50(1.08) = $229.50, choice C.\"},{\"prompt\":\"The percentage of surveyed students who prefer online homework increased from 40% to 52%. By what percent did the percentage increase?\",\"options\":[\"12%\",\"20%\",\"30%\",\"52%\"],\"answer\":2,\"drill\":\"Drill 2\",\"hint\":\"Find the increase in percentage points, then divide that increase by the original 40%.\",\"explanation\":\"The increase is 52 \\u2212 40 = 12 percentage points. Relative to the original percentage, the percent increase is 12\\\/40 \\u00d7 100% = 30%, choice C.\"},{\"prompt\":\"If 24 is x percent of 15, what is x?\",\"options\":[\"1.6\",\"9\",\"62.5\",\"160\"],\"answer\":3,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"An investment of $18,000 earns 4.5% simple interest in one year. What is the value of the investment after one year?\",\"options\":[\"$810\",\"$18,045\",\"$18,450\",\"$18,810\"],\"answer\":3,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"A quantity is increased by 25% and then the new value is decreased by 20%. Compared with the original quantity, what is the final percentage change?\",\"options\":[\"A 5% decrease\",\"No change\",\"A 5% increase\",\"A 10% increase\"],\"answer\":1,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"On a map, 1 inch represents 18 miles. Two towns are 2.75 inches apart on the map. What is their actual distance?\",\"options\":[\"20.75 miles\",\"36 miles\",\"49.5 miles\",\"54 miles\"],\"answer\":2,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"The percentage of surveyed students who prefer online homework increased from 40% to 52%. By what percent did the percentage increase?\",\"options\":[\"12%\",\"20%\",\"30%\",\"52%\"],\"answer\":2,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A price of $100 increases by 20% and then decreases by 20%. What is the final price?\",\"options\":[\"$80\",\"$96\",\"$100\",\"$104\"],\"answer\":1,\"drill\":\"Common Mistakes\",\"hint\":\"Apply each percent change to the current price.\",\"explanation\":\"After the increase, the price is $120. A 20% decrease is $24, so the final price is $96. Equal percentage increases and decreases do not cancel because they use different bases.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A $250 tablet is discounted by 15%, and then 8% sales tax is applied to the discounted price. What is the final price?\",\"options\":[\"$212.50\",\"$227.50\",\"$229.50\",\"$232.00\"],\"answer\":2,\"drill\":\"Challenge Question\",\"hint\":\"Apply the discount first, then apply tax to the discounted price.\",\"explanation\":\"$250(0.85) = $212.50. Then $212.50(1.08) = $229.50.\"}]},{\"title\":\"Coordinate Geometry\",\"questions\":[{\"prompt\":\"A line has slope \\u22123\\\/2 and passes through the point (0, \\u22125). Which equation defines the line?\",\"options\":[\"g(x) = \\u2212(3\\\/2)x \\u2212 5\",\"g(x) = \\u2212(3\\\/2)x + 5\",\"g(x) = (3\\\/2)x \\u2212 5\",\"g(x) = (3\\\/2)x + 5\"],\"answer\":0,\"drill\":\"Drill 1\",\"hint\":\"Use slope-intercept form y = mx + b. The point (0, \\u22125) gives the y-intercept directly.\",\"explanation\":\"The slope is \\u22123\\\/2, so m = \\u22123\\\/2. Because the line passes through (0, \\u22125), b = \\u22125. Therefore g(x) = \\u2212(3\\\/2)x \\u2212 5, choice A.\"},{\"prompt\":\"What is the value of a, the x-coordinate of the x-intercept?\",\"options\":[\"\\u22128\",\"\\u22125\",\"\\u22124\",\"0\"],\"answer\":1,\"drill\":\"Drill 1\",\"hint\":\"An x-intercept is where the graph touches or crosses the x-axis, so y = 0.\",\"explanation\":\"The parabola touches the x-axis at its vertex. Reading the grid, that point is (\\u22125, 0), so a = \\u22125.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/coordinate-geometry\\\/redrawn\\\/parabola-x-intercept.svg\"},{\"prompt\":\"Lines m and n are parallel. Line n is defined by 12x = \\u22129y + 18. What is the slope of line m?\",\"options\":[\"\\u22124\\\/3\",\"\\u22123\\\/4\",\"3\\\/4\",\"4\\\/3\"],\"answer\":0,\"drill\":\"Drill 1\",\"hint\":\"Rewrite the equation of line n in y = mx + b form. Parallel lines have equal slopes.\",\"explanation\":\"From 12x = \\u22129y + 18, add 9y and subtract 12x: 9y = \\u221212x + 18. Thus y = \\u2212(4\\\/3)x + 2. Line m is parallel to n, so its slope is \\u22124\\\/3.\"},{\"prompt\":\"The function g is defined by g(x) = \\u22123x\\u00b2 \\u2212 18x \\u2212 15. For what value of x does g(x) reach its maximum?\",\"options\":[\"\\u22125\",\"\\u22123\",\"\\u22121\",\"12\"],\"answer\":1,\"drill\":\"Drill 1\",\"hint\":\"The maximum occurs at the vertex. For ax\\u00b2 + bx + c, its x-coordinate is \\u2212b\\\/(2a).\",\"explanation\":\"Here a = \\u22123 and b = \\u221218. The vertex x-coordinate is \\u2212(\\u221218)\\\/(2\\u00b7\\u22123) = 18\\\/\\u22126 = \\u22123. Since a is negative, the parabola opens downward, so this is the maximum.\"},{\"prompt\":\"The system 3x\\u00b2 + y = 3 and 4x \\u2212 y = 1 has two solutions. Which value is the y-coordinate of one solution?\",\"options\":[\"2\\\/3\",\"1\",\"5\\\/3\",\"9\"],\"answer\":2,\"drill\":\"Drill 1\",\"hint\":\"Solve the linear equation for y, substitute it into the quadratic equation, then find y from either x-value.\",\"explanation\":\"From 4x \\u2212 y = 1, y = 4x \\u2212 1. Substitute: 3x\\u00b2 + 4x \\u2212 1 = 3, so 3x\\u00b2 + 4x \\u2212 4 = 0 = (3x \\u2212 2)(x + 2). For x = 2\\\/3, y = 4(2\\\/3) \\u2212 1 = 5\\\/3, which is choice C.\"},{\"prompt\":\"The point (1, y) lies on the line. What is y?\",\"options\":[\"\\u22121\\\/2\",\"\\u22122\\\/5\",\"\\u22121\\\/5\",\"5\\\/2\"],\"answer\":0,\"drill\":\"Drill 2\",\"hint\":\"Use two clear points on the graph to find the slope, then write the line equation.\",\"explanation\":\"The graph passes through (0, \\u22123) and (2, 2), so the slope is (2 \\u2212 (\\u22123))\\\/(2 \\u2212 0) = 5\\\/2. Thus y = (5\\\/2)x \\u2212 3. At x = 1, y = 5\\\/2 \\u2212 3 = \\u22121\\\/2, choice A.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/coordinate-geometry\\\/redrawn\\\/rising-line.svg\"},{\"prompt\":\"Line \\u2113 is defined by y = 5x \\u2212 8. Line m is parallel to line \\u2113. Which equation could represent line m?\",\"options\":[\"\\u22125x + y = 11\",\"\\u22123x + 5y = 15\",\"x + 5y = \\u22122\",\"10x + 2y = 9\"],\"answer\":0,\"drill\":\"Drill 2\",\"hint\":\"Parallel lines have equal slopes. Rewrite each option in y = mx + b form.\",\"explanation\":\"Line l has slope 5. Choice A becomes y = 5x + 11, which also has slope 5 but a different intercept. Therefore it can represent the parallel line m.\"},{\"prompt\":\"The equation y = 36x\\u00b2 \\u2212 48x + c has no real x-intercepts. Which is a possible value of c?\",\"options\":[\"\\u22124\",\"9\",\"16\",\"24\"],\"answer\":3,\"drill\":\"Drill 2\",\"hint\":\"For ax\\u00b2 + bx + c = 0 to have no real roots, its discriminant b\\u00b2 \\u2212 4ac must be negative.\",\"explanation\":\"The discriminant is (\\u221248)\\u00b2 \\u2212 4(36)c = 2304 \\u2212 144c. It is negative when c \\u003E 16. Of the choices, only 24 meets that condition, so D is correct.\"},{\"prompt\":\"The table contains the points (6, \\u221227), (11, \\u221252), and (20, \\u221297) for the linear function f(x) = mx + 3. What is the value of m?\",\"options\":[\"\\u22127\",\"\\u22125\",\"5\",\"7\"],\"answer\":1,\"drill\":\"Drill 2\",\"hint\":\"Substitute any row from the table into f(x) = mx + 3.\",\"explanation\":\"Using x = 6 and f(x) = \\u221227 gives \\u221227 = 6m + 3. Then 6m = \\u221230, so m = \\u22125. The other rows confirm the same slope.\"},{\"prompt\":\"The graph of y = k(x \\u2212 9)(x + 3) is a parabola with vertex (a, b). Which expression is equal to b?\",\"options\":[\"\\u221248k\",\"\\u221236k\",\"\\u221227k\",\"\\u22123k\"],\"answer\":1,\"drill\":\"Drill 2\",\"hint\":\"The x-coordinate of the vertex lies halfway between the two roots.\",\"explanation\":\"The roots are 9 and \\u22123, so the axis of symmetry is x = (9 + (\\u22123))\\\/2 = 3. Substitute x = 3: b = k(3 \\u2212 9)(3 + 3) = k(\\u22126)(6) = \\u221236k, choice B.\"},{\"prompt\":\"What is a if the line passes through (a, 5)?\",\"options\":[\"12\\\/7\",\"14\\\/3\",\"21\\\/4\",\"88\\\/7\"],\"answer\":1,\"drill\":\"Drill 3\",\"hint\":\"Read two points from the graph, find the slope, and use the line equation with y = 5.\",\"explanation\":\"The line passes through (0, 3) and (7, 6), so its slope is 3\\\/7 and y = (3\\\/7)x + 3. Set y = 5: 5 = (3\\\/7)a + 3, so 2 = 3a\\\/7 and a = 14\\\/3, choice B.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/coordinate-geometry\\\/redrawn\\\/gentle-rising-line.svg\"},{\"prompt\":\"A line has slope \\u22124 and passes through the point (0, 5). Which set of ordered pairs lies on the line?\",\"options\":[\"(\\u22121, 1), (0, 5), (1, 9)\",\"(\\u22121, 9), (0, 5), (1, 1)\",\"(\\u22121, 5), (0, 1), (1, \\u22123)\",\"(\\u22121, 9), (0, 4), (1, 1)\"],\"answer\":1,\"drill\":\"Drill 3\",\"hint\":\"Write the equation y = \\u22124x + 5 and substitute x = \\u22121, 0, and 1.\",\"explanation\":\"For y = \\u22124x + 5, the corresponding values are 9, 5, and 1 when x is \\u22121, 0, and 1. These form (\\u22121, 9), (0, 5), and (1, 1), choice B.\"},{\"prompt\":\"A straight tunnel on a coordinate plan follows 4x + 7y = 28. A ventilation shaft is perpendicular to the tunnel. What slope must the shaft have on this plan?\",\"options\":[\"\\u22127\\\/4\",\"\\u22124\\\/7\",\"4\\\/7\",\"7\\\/4\"],\"answer\":3,\"drill\":\"Drill 3\",\"hint\":\"First identify the tunnel slope.\",\"explanation\":\"The tunnel equation becomes y = (\\u22124\\\/7)x + 4. The negative reciprocal of \\u22124\\\/7 is 7\\\/4, the shaft slope.\"},{\"prompt\":\"A line through the origin contains the points (p, \\u22123) and (\\u221227, p). Which is a possible value of p?\",\"options\":[\"\\u221218\",\"\\u22129\",\"\\u22123\",\"0\"],\"answer\":1,\"drill\":\"Drill 3\",\"hint\":\"Because the line passes through the origin, each point has the same ratio y\\\/x.\",\"explanation\":\"The slopes from the origin are \\u22123\\\/p and p\\\/(\\u221227). Set them equal: \\u22123\\\/p = \\u2212p\\\/27. This gives p\\u00b2 = 81, so p = \\u00b19. The listed possible value is \\u22129, choice B.\"},{\"prompt\":\"The function f is defined by f(x) = \\u22123x \\u2212 42. Its graph has x-intercept (c, 0) and y-intercept (0, d). What is d\\\/c?\",\"options\":[\"\\u221214\",\"\\u22123\",\"3\",\"14\"],\"answer\":2,\"drill\":\"Drill 3\",\"hint\":\"Find each intercept by setting the other coordinate equal to zero.\",\"explanation\":\"For the x-intercept, 0 = \\u22123x \\u2212 42 gives c = \\u221214. At x = 0, d = \\u221242. Therefore d\\\/c = (\\u221242)\\\/(\\u221214) = 3, choice C.\"},{\"prompt\":\"A line contains the points (33, p) and (21, p \\u2212 4). Its x-intercept is (a, p + 3). What is the value of a?\",\"options\":[\"24\",\"33\",\"42\",\"45\"],\"answer\":2,\"drill\":\"Drill 3\",\"hint\":\"An x-intercept has y-coordinate 0, so first use p + 3 = 0.\",\"explanation\":\"Since (a, p + 3) is an x-intercept, p + 3 = 0 and p = \\u22123. The two table points have slope ((p\\u22124)\\u2212p)\\\/(21\\u221233) = \\u22124\\\/\\u221212 = 1\\\/3. From (33, \\u22123) to y = 0, y rises 3, so x rises 9. Thus a = 33 + 9 = 42.\"},{\"prompt\":\"The graph shows a straight route through the labeled points (0, 2) and (8, 6). A checkpoint on this route has y-coordinate 9. What is its x-coordinate?\",\"options\":[\"12\",\"14\",\"16\",\"18\"],\"answer\":1,\"drill\":\"Drill 4\",\"hint\":\"Find the slope using the two labeled points, then solve for x at y = 9.\",\"explanation\":\"The slope is (6 \\u2212 2)\\\/(8 \\u2212 0) = 1\\\/2, so y = x\\\/2 + 2. Setting y = 9 gives x\\\/2 = 7 and x = 14.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/coordinate-geometry\\\/redrawn\\\/original-route-line.svg\"},{\"prompt\":\"Which set of points matches the graph?\",\"options\":[\"(\\u22121, \\u22123), (0, 3), (1, 9)\",\"(\\u22121, 3), (0, \\u22123), (1, \\u22129)\",\"(\\u22121, 9), (0, 3), (1, \\u22123)\",\"(\\u22121, \\u22129), (0, \\u22123), (1, 3)\"],\"answer\":2,\"drill\":\"Drill 4\",\"hint\":\"Read the graph at x = \\u22121, 0, and 1.\",\"explanation\":\"The corresponding points are (\\u22121, 9), (0, 3), and (1, \\u22123). These values are listed in choice C.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/coordinate-geometry\\\/redrawn\\\/falling-line.svg\"},{\"prompt\":\"On a design grid, a wall lies along 9x \\u2212 5y = 45. A support must pass through (3, 4) and meet the wall at a right angle. Which equation describes the support?\",\"options\":[\"9x \\u2212 5y = 7\",\"5x + 9y = 51\",\"5x \\u2212 9y = 51\",\"9x + 5y = 47\"],\"answer\":1,\"drill\":\"Drill 4\",\"hint\":\"The support must have both the perpendicular slope and the specified point.\",\"explanation\":\"The wall slope is 9\\\/5, so the support slope is \\u22125\\\/9. The equation 5x + 9y = 51 has this slope and is satisfied by (3, 4), since 15 + 36 = 51.\"},{\"prompt\":\"Two markers on a straight survey line through the origin are at (k, \\u22128) and (\\u221218, k). If k is positive, what is its value?\",\"options\":[\"6\",\"12\",\"18\",\"24\"],\"answer\":1,\"drill\":\"Drill 4\",\"hint\":\"The two slopes measured from the origin must agree.\",\"explanation\":\"Since the origin and both markers are collinear, \\u22128\\\/k = k\\\/(\\u221218). Thus k\\u00b2 = 144. The positive solution is k = 12.\"},{\"prompt\":\"A triangular sign occupies the first-quadrant region bounded by the axes and the line y = \\u22122x + 18. What is its area, in square units?\",\"options\":[\"54\",\"81\",\"162\",\"324\"],\"answer\":1,\"drill\":\"Drill 4\",\"hint\":\"Use the intercepts as the lengths of the perpendicular sides.\",\"explanation\":\"At x = 0 the line has y = 18; at y = 0 it has x = 9. The right triangle therefore has area (1\\\/2)(9)(18) = 81 square units.\"},{\"prompt\":\"A straight pipe on a coordinate plan passes through (5, r) and (17, r + 9). It crosses the x-axis at (a, r \\u2212 6). What is a?\",\"options\":[\"\\u22128\",\"\\u22123\",\"3\",\"13\"],\"answer\":1,\"drill\":\"Drill 4\",\"hint\":\"An x-intercept has y-coordinate zero; determine r before finding the line equation.\",\"explanation\":\"At the intercept r \\u2212 6 = 0, so r = 6. The slope is 9\\\/12 = 3\\\/4. Using (5, 6), set y = 0 in y \\u2212 6 = (3\\\/4)(x \\u2212 5). Then x \\u2212 5 = \\u22128, so a = \\u22123.\"},{\"prompt\":\"The equations 4a \\u2212 3b = 14 and a + 3b = 6 form a system. What is the value of a?\",\"options\":[\"2\",\"4\",\"8\",\"20\"],\"answer\":1,\"drill\":\"Drill 5\",\"hint\":\"Add the equations so the b terms cancel.\",\"explanation\":\"Adding 4a \\u2212 3b = 14 and a + 3b = 6 gives 5a = 20. Therefore a = 4, choice B.\"},{\"prompt\":\"The graph of y = 2x\\u00b2 \\u2212 15x intersects the horizontal line y = b at exactly one point, (a, b). What is a?\",\"options\":[\"15\\\/8\",\"15\\\/4\",\"4\",\"15\\\/2\"],\"answer\":1,\"drill\":\"Drill 5\",\"hint\":\"A horizontal line intersects a parabola once only when it passes through the vertex.\",\"explanation\":\"For y = 2x\\u00b2 \\u2212 15x, the vertex x-coordinate is \\u2212b\\\/(2a) = \\u2212(\\u221215)\\\/(2\\u00b72) = 15\\\/4. Thus a = 15\\\/4.\"},{\"prompt\":\"The graphs y = 4x + 20 and y = \\u2212(1\\\/2)x\\u00b2 + 44 intersect at (x, y). If x \\u003E 0, what is x?\",\"options\":[\"1\\\/2\",\"2\",\"4\",\"12\"],\"answer\":2,\"drill\":\"Drill 5\",\"hint\":\"Rewrite the first equation as y = 4x + 20, then set the two y-expressions equal.\",\"explanation\":\"Equating 4x + 20 and \\u2212(1\\\/2)x\\u00b2 + 44 gives x\\u00b2 + 8x \\u2212 48 = 0, or (x + 12)(x \\u2212 4) = 0. Since x \\u003E 0, x = 4, choice C.\"},{\"prompt\":\"For what value of a does the equation (2\\\/5)(15x \\u2212 5) = a \\u2212 3ax have infinitely many solutions?\",\"options\":[\"\\u22125\",\"\\u22125\\\/2\",\"\\u22122\",\"\\u22122\\\/5\"],\"answer\":2,\"drill\":\"Drill 5\",\"hint\":\"For infinitely many solutions, both sides must simplify to the identical expression.\",\"explanation\":\"The left side simplifies to (2\\\/5)(15x \\u2212 5) = 6x \\u2212 2. The right side is a \\u2212 3ax. Matching the constant term gives a = \\u22122, and then \\u22123a = 6 also holds. Choice C is correct.\"},{\"prompt\":\"The graphs y = 4x\\u00b2 + 15x and y = \\u2212x intersect at the origin and at the point (\\u2212b, b). What is b\\u00b2?\",\"options\":[\"4\",\"8\",\"16\",\"25\"],\"answer\":2,\"drill\":\"Drill 5\",\"hint\":\"Set the two equations equal and solve for the nonzero intersection x-coordinate.\",\"explanation\":\"Set 4x\\u00b2 + 15x = \\u2212x: 4x\\u00b2 + 16x = 0, so 4x(x + 4) = 0. The second intersection has x = \\u22124. Since it is written (\\u2212b, b), b = 4 and b\\u00b2 = 16.\"},{\"prompt\":\"A polynomial graph crosses the x-axis at x = \\u22122 and x = 5 and touches, but does not cross, the x-axis at x = 1. Which equation could represent the graph?\",\"options\":[\"y = (x + 2)(x \\u2212 1)(x \\u2212 5)\",\"y = (x \\u2212 2)(x + 1)\\u00b2(x + 5)\",\"y = (x + 2)(x \\u2212 1)\\u00b2(x \\u2212 5)\",\"y = (x + 2)\\u00b2(x \\u2212 1)(x \\u2212 5)\\u00b2\"],\"answer\":2,\"drill\":\"Drill 6\",\"hint\":\"A graph touches the x-axis at a root with even multiplicity and crosses at roots with odd multiplicity.\",\"explanation\":\"The factors for crossings at \\u22122 and 5 are (x + 2) and (x \\u2212 5). Touching at x = 1 requires an even power, such as (x \\u2212 1)\\u00b2. Thus y = (x + 2)(x \\u2212 1)\\u00b2(x \\u2212 5), choice C.\"},{\"prompt\":\"The equations y = 4x\\u00b2 \\u2212 4x + 3 and y = 3x + 2 are graphed in the coordinate plane. At how many points do the graphs intersect?\",\"options\":[\"0\",\"1\",\"2\",\"3\"],\"answer\":2,\"drill\":\"Drill 6\",\"hint\":\"Set the two expressions for y equal and examine the discriminant.\",\"explanation\":\"Equating them gives 4x\\u00b2 \\u2212 4x + 3 = 3x + 2, or 4x\\u00b2 \\u2212 7x + 1 = 0. Its discriminant is 49 \\u2212 16 = 33, which is positive, so there are two distinct real intersections.\"},{\"prompt\":\"The function p is defined by p(x) = \\u22123x\\u00b2 + 24x \\u2212 41, and q(x) = p(x \\u2212 2). At what x-value does q(x) reach its maximum?\",\"options\":[\"2\",\"4\",\"6\",\"7\"],\"answer\":2,\"drill\":\"Drill 6\",\"hint\":\"Find the vertex input of p, then account for the horizontal shift in p(x \\u2212 2).\",\"explanation\":\"The vertex of p(x) = \\u22123x\\u00b2 + 24x \\u2212 41 occurs at x = \\u221224\\\/(2\\u00b7\\u22123) = 4. For q(x) = p(x \\u2212 2), the input x \\u2212 2 must equal 4, so x = 6, choice C.\"},{\"prompt\":\"The function g is defined by g(x) = (x \\u2212 8)(x \\u2212 7)(x \\u2212 4), and h(x) = g(x + 3). What is the sum of the three x-intercepts of h?\",\"options\":[\"\\u221228\",\"\\u221216\",\"10\",\"28\"],\"answer\":2,\"drill\":\"Drill 6\",\"hint\":\"Set each factor of g(x + 3) equal to zero.\",\"explanation\":\"The roots of g are 8, 7, and 4. For h(x) = g(x + 3), the roots shift left 3 units to 5, 4, and 1. Their sum is 10, choice C.\"},{\"prompt\":\"The graphs y = (x \\u2212 5)\\u00b2 and y = 8x \\u2212 56 intersect at exactly one point, (a, b). What is a + b?\",\"options\":[\"\\u221211\",\"12\",\"15\",\"25\"],\"answer\":3,\"drill\":\"Drill 6\",\"hint\":\"Set the parabola equal to the line and solve for x.\",\"explanation\":\"Set (x \\u2212 5)\\u00b2 = 8x \\u2212 56. Expanding gives x\\u00b2 \\u2212 18x + 81 = 0, or (x \\u2212 9)\\u00b2 = 0, so a = 9. Then b = 8(9) \\u2212 56 = 16. Therefore a + b = 25, choice D.\"},{\"prompt\":\"A line has slope \\u22123\\\/2 and passes through the point (0, \\u22125). Which equation defines the line?\",\"options\":[\"g(x) = \\u2212(3\\\/2)x \\u2212 5\",\"g(x) = \\u2212(3\\\/2)x + 5\",\"g(x) = (3\\\/2)x \\u2212 5\",\"g(x) = (3\\\/2)x + 5\"],\"answer\":0,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"The table contains the points (6, \\u221227), (11, \\u221252), and (20, \\u221297) for the linear function f(x) = mx + 3. What is the value of m?\",\"options\":[\"\\u22127\",\"\\u22125\",\"5\",\"7\"],\"answer\":1,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"The graph shows a straight route through the labeled points (0, 2) and (8, 6). A checkpoint on this route has y-coordinate 9. What is its x-coordinate?\",\"options\":[\"12\",\"14\",\"16\",\"18\"],\"answer\":1,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/coordinate-geometry\\\/redrawn\\\/original-route-line.svg\"},{\"prompt\":\"The graph of y = 2x\\u00b2 \\u2212 15x intersects the horizontal line y = b at exactly one point, (a, b). What is a?\",\"options\":[\"15\\\/8\",\"15\\\/4\",\"4\",\"15\\\/2\"],\"answer\":1,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"The graphs y = (x \\u2212 5)\\u00b2 and y = 8x \\u2212 56 intersect at exactly one point, (a, b). What is a + b?\",\"options\":[\"\\u221211\",\"12\",\"15\",\"25\"],\"answer\":3,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"What is the slope of the line through (\\u22122, 5) and (4, \\u22127)?\",\"options\":[\"\\u22122\",\"\\u22121\\\/2\",\"1\\\/2\",\"2\"],\"answer\":0,\"drill\":\"Common Mistakes\",\"hint\":\"Subtract coordinates in the same order in the numerator and denominator.\",\"explanation\":\"The slope is (\\u22127 \\u2212 5)\\\/(4 \\u2212 (\\u22122)) = \\u221212\\\/6 = \\u22122.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"The endpoints of a diameter of a circle are (\\u22122, 3) and (6, 3). Which equation represents the circle?\",\"options\":[\"(x \\u2212 2)\\u00b2 + (y \\u2212 3)\\u00b2 = 16\",\"(x + 2)\\u00b2 + (y \\u2212 3)\\u00b2 = 16\",\"(x \\u2212 2)\\u00b2 + (y \\u2212 3)\\u00b2 = 64\",\"(x \\u2212 4)\\u00b2 + y\\u00b2 = 16\"],\"answer\":0,\"drill\":\"Challenge Question\",\"hint\":\"Find the midpoint and half the diameter.\",\"explanation\":\"The center is the midpoint (2, 3), and the diameter is 8, so the radius is 4. The equation is (x \\u2212 2)\\u00b2 + (y \\u2212 3)\\u00b2 = 16.\"}]},{\"title\":\"Data Analysis\",\"questions\":[{\"prompt\":\"What is the average annual temperature of the coldest state shown?\",\"options\":[\"20\\u00b0F\",\"30\\u00b0F\",\"40\\u00b0F\",\"50\\u00b0F\"],\"answer\":1,\"drill\":\"Drill 1\",\"hint\":\"Find the shortest bar and read its height from the vertical scale.\",\"explanation\":\"State 2 has the shortest bar, and its height aligns with 30\\u00b0F. Therefore the coldest state\\u0027s average annual temperature is 30\\u00b0F.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/data-analysis\\\/redrawn\\\/state-temperatures.svg\"},{\"prompt\":\"On which day did the runner take the fewest breaks?\",\"options\":[\"Monday\",\"Wednesday\",\"Thursday\",\"Friday\"],\"answer\":2,\"drill\":\"Drill 1\",\"hint\":\"Find the lowest point on the line graph.\",\"explanation\":\"The graph shows 5 breaks Monday, 3 Tuesday, 4 Wednesday, 1 Thursday, and 2 Friday. The minimum is 1 break on Thursday, choice C.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/data-analysis\\\/redrawn\\\/runner-breaks.svg\"},{\"prompt\":\"According to the line of best fit, what paycheck is predicted for 26 hours of work?\",\"options\":[\"$260\",\"$279\",\"$289\",\"$297\"],\"answer\":3,\"drill\":\"Drill 1\",\"hint\":\"Locate 26 hours on the horizontal axis and read the line of best fit, not an individual data point.\",\"explanation\":\"At x = 26 hours, the line of best fit is just below $300, at approximately $297. Therefore choice D is the best estimate.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/data-analysis\\\/redrawn\\\/paycheck-scatterplot.svg\"},{\"prompt\":\"A greenhouse inventory lists annual plants as 6,700 spring blooming, 2,500 summer blooming, and 1,200 fall blooming; perennial plants as 3,200 spring blooming, 3,500 summer blooming, and 5,300 fall blooming. What proportion of the perennial plants are summer blooming?\",\"options\":[\"5\\\/32\",\"25\\\/224\",\"7\\\/24\",\"7\\\/12\"],\"answer\":2,\"drill\":\"Drill 1\",\"hint\":\"Use summer-blooming perennials over all perennials.\",\"explanation\":\"There are 3,500 summer-blooming perennials out of 12,000 perennials total. The proportion is 3,500\\\/12,000 = 35\\\/120 = 7\\\/24, choice C.\"},{\"prompt\":\"What is the best interpretation of the number 20 in the box-plot data?\",\"options\":[\"The highest number of points\",\"The range of points\",\"The mean number of points\",\"The median number of points\"],\"answer\":1,\"drill\":\"Drill 1\",\"hint\":\"Subtract the minimum value at the left whisker from the maximum at the right whisker.\",\"explanation\":\"The minimum is 4 points and the maximum is 24 points. Their difference is 24 \\u2212 4 = 20, so 20 is the range of points scored, choice B.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/data-analysis\\\/redrawn\\\/points-boxplot.svg\"},{\"prompt\":\"A coat costs $48 after a 20% discount. What was its original price?\",\"options\":[\"$38.40\",\"$57.60\",\"$60.00\",\"$72.00\"],\"answer\":2,\"drill\":\"Drill 2\",\"hint\":\"After a 20% discount, the sale price is 80% of the original price.\",\"explanation\":\"Let p be the original price. Then 0.80p = 48, so p = 48\\\/0.80 = 60. The original price was $60.00, choice C.\"},{\"prompt\":\"A retirement account decreased by 3%. The current amount is a times last year\\u0027s amount. What is a?\",\"options\":[\"0.03\",\"0.97\",\"1.03\",\"1.97\"],\"answer\":1,\"drill\":\"Drill 2\",\"hint\":\"Subtract the decrease rate from 1.\",\"explanation\":\"After a 3% decrease, the account retains 100% \\u2212 3% = 97% of its former value. As a decimal multiplier, a = 0.97, choice B.\"},{\"prompt\":\"At 6 megabytes per second, approximately how many minutes are needed to download sixteen 110-megabyte files?\",\"options\":[\"1\",\"5\",\"50\",\"300\"],\"answer\":1,\"drill\":\"Drill 2\",\"hint\":\"Find the total megabytes, divide by the rate to get seconds, then divide by 60.\",\"explanation\":\"The files total 16\\u00b7110 = 1,760 megabytes. At 6 megabytes per second, the time is 1,760\\\/6 \\u2248 293.3 seconds. Dividing by 60 gives about 4.9 minutes, approximately 5 minutes, choice B.\"},{\"prompt\":\"A library surveys students about their preferred workshop. Among Grade 9 students, 42 choose coding and 28 choose design. The respective counts are 54 and 36 in Grade 10, 63 and 21 in Grade 11, and 48 and 32 in Grade 12. Which grade has the greatest ratio of coding to design preferences?\",\"options\":[\"Grade 9\",\"Grade 10\",\"Grade 11\",\"Grade 12\"],\"answer\":2,\"drill\":\"Drill 2\",\"hint\":\"Compare the coding count divided by the design count for each grade.\",\"explanation\":\"The ratios are 42\\\/28 = 1.5, 54\\\/36 = 1.5, 63\\\/21 = 3, and 48\\\/32 = 1.5. Grade 11 has the greatest ratio.\"},{\"prompt\":\"Estimate how many employees across 30 offices have fewer than two dogs, given 18 employees per office and 27.8% have at least two dogs.\",\"options\":[\"150\",\"390\",\"540\",\"2,700\"],\"answer\":1,\"drill\":\"Drill 2\",\"hint\":\"Find the total employees, then use the complement of 27.8%.\",\"explanation\":\"The company has about 30\\u00b718 = 540 employees. If 27.8% have at least two dogs, then 72.2% have fewer than two. Thus 0.722(540) = 389.88, which is about 390 employees, choice B.\"},{\"prompt\":\"What is the median of 343, 272, 321, 283, 272, 438, and 309?\",\"options\":[\"283\",\"309\",\"321\",\"343\"],\"answer\":1,\"drill\":\"Drill 3\",\"hint\":\"Order all seven values and choose the fourth one.\",\"explanation\":\"In order, the values are 272, 272, 283, 309, 321, 343, 438. With seven values, the median is the fourth value, 309.\"},{\"prompt\":\"Data set P has a minimum of 13 and a maximum of 30. Data set Q has a minimum of 2 and a maximum of 19. How do their ranges compare?\",\"options\":[\"The ranges are equal\",\"P\\u0027s range is less\",\"P\\u0027s range is greater\",\"There is not enough information\"],\"answer\":0,\"drill\":\"Drill 3\",\"hint\":\"For each set, subtract its minimum from its maximum.\",\"explanation\":\"For P, the range is 30 \\u2212 13 = 17. For Q, the range is 19 \\u2212 2 = 17. The ranges are equal, choice A.\"},{\"prompt\":\"A stem-and-leaf plot represents the values 7, 12, 16, 23, 26, 26, 32, 36, and 40. Which value is closest to their mean?\",\"options\":[\"3.4\",\"6.2\",\"24.2\",\"26.0\"],\"answer\":2,\"drill\":\"Drill 3\",\"hint\":\"Translate the plot into 7, 12, 16, 23, 26, 26, 32, 36, and 40.\",\"explanation\":\"The nine values sum to 218. Their mean is 218\\\/9 \\u2248 24.22, which is closest to 24.2, choice C.\"},{\"prompt\":\"Which statement must be true based on the two company box plots?\",\"options\":[\"Company A\\u0027s median is 5 years younger\",\"Company A\\u0027s median is 5 years older\",\"Company B has the greater range\",\"Both companies have the same range\"],\"answer\":0,\"drill\":\"Drill 3\",\"hint\":\"Read the vertical median lines inside the boxes against the age scale.\",\"explanation\":\"Company A\\u0027s median is 35 years and Company B\\u0027s median is 40 years. Therefore Company A\\u0027s median age is 5 years younger than Company B\\u0027s, choice A.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/data-analysis\\\/redrawn\\\/company-boxplots.svg\"},{\"prompt\":\"A graph shows the current ages of a group of students. A second graph is made using the ages of the same students exactly 5 years ago. How do the means and ranges compare?\",\"options\":[\"Equal means; current range smaller\",\"Equal means; current range greater\",\"Current mean smaller; same range\",\"Current mean greater; same range\"],\"answer\":3,\"drill\":\"Drill 3\",\"hint\":\"Subtracting the same number from every data value changes the center but not the spread from minimum to maximum.\",\"explanation\":\"Every age five years ago is exactly 5 less than the corresponding current age, so the earlier mean is 5 less and the current mean is greater. Subtracting 5 from both the minimum and maximum leaves their difference unchanged, so the ranges are equal. Choice D is correct.\"},{\"prompt\":\"A frequency table lists the value 2 with frequency 3, the value 3 with frequency 1, and the value 6 with frequency 4. Which data set correctly represents the table?\",\"options\":[\"1, 1, 1, 3, 3, 4, 4, 4, 4, 4, 4\",\"2, 2, 2, 3, 6, 6, 6, 6\",\"3, 6, 24\",\"1, 2, 3, 3, 4, 6\"],\"answer\":1,\"drill\":\"Drill 4\",\"hint\":\"Write each number as many times as its listed frequency.\",\"explanation\":\"The table lists 2 with frequency 3, 3 with frequency 1, and 6 with frequency 4. Therefore the data set is 2, 2, 2, 3, 6, 6, 6, 6, which is choice B.\"},{\"prompt\":\"The graph shows the number of reusable bottles filled by a school station each day. How many more bottles were filled on Thursday than on Tuesday?\",\"options\":[\"20\",\"30\",\"40\",\"60\"],\"answer\":2,\"drill\":\"Drill 4\",\"hint\":\"Read both daily totals and subtract Tuesday from Thursday.\",\"explanation\":\"Thursday has 190 bottles and Tuesday has 150 bottles, so the difference is 190 \\u2212 150 = 40 bottles.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/data-analysis\\\/redrawn\\\/original-refill-linegraph.svg\"},{\"prompt\":\"What was the household\\u0027s water usage, in gallons, during the month in which it used the most water?\",\"options\":[\"5\",\"1,000\",\"5,000\",\"27,000\"],\"answer\":2,\"drill\":\"Drill 4\",\"hint\":\"Find the tallest bar, then account for the vertical axis being measured in thousands of gallons.\",\"explanation\":\"The tallest bar occurs in July and reaches 5 on an axis labeled thousands of gallons. Thus the household used 5 \\u00d7 1,000 = 5,000 gallons, choice C.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/data-analysis\\\/redrawn\\\/water-usage.svg\"},{\"prompt\":\"Using the line of best fit, what is the predicted test score for a student who studies 1.5 hours?\",\"options\":[\"56\",\"64\",\"68\",\"72\"],\"answer\":1,\"drill\":\"Drill 4\",\"hint\":\"Move up from 1.5 hours to the line of best fit, then read the corresponding test score.\",\"explanation\":\"At 1.5 hours, the line of best fit is near a test score of 64. Therefore 64, choice B, is the closest prediction.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/data-analysis\\\/redrawn\\\/study-scatterplot.svg\"},{\"prompt\":\"The data values are 12, 15, 15, 18, 20, 20, 20, 24, and 27. Which statement is true?\",\"options\":[\"The mode is 15\",\"The median is 18\",\"The mean is 20\",\"The mode is 20\"],\"answer\":3,\"drill\":\"Drill 4\",\"hint\":\"The mode is the value that occurs most often.\",\"explanation\":\"The value 20 appears three times, more often than any other value. Therefore the mode is 20, choice D. The median is also 20, not 18.\"},{\"prompt\":\"A box plot has minimum 14, first quartile 22, median 31, third quartile 38, and maximum 50. Which statement must be true?\",\"options\":[\"The range is 28\",\"The interquartile range is 16\",\"The mean is 31\",\"The mode is 22\"],\"answer\":1,\"drill\":\"Drill 4\",\"hint\":\"The interquartile range equals the third quartile minus the first quartile.\",\"explanation\":\"The interquartile range is 38 \\u2212 22 = 16, so choice B must be true. The range is 50 \\u2212 14 = 36, and a box plot does not determine the mean or mode.\"},{\"prompt\":\"The orbital periods of eight planets, in Earth years, are 0.24, 0.62, 1.00, 1.88, 11.86, 29.46, 84.07, and 164.95. What is their average orbital period?\",\"options\":[\"29.48\",\"36.76\",\"42.01\",\"294.08\"],\"answer\":1,\"drill\":\"Drill 5\",\"hint\":\"Add all eight orbital periods and divide the total by 8.\",\"explanation\":\"The orbital periods sum to 294.08 Earth years. Dividing by 8 gives 294.08 \\u00f7 8 = 36.76 Earth years, choice B.\"},{\"prompt\":\"The heights of five children, in inches, are 47, 51, 53, 61, and 63. What is the median height?\",\"options\":[\"47\",\"52\",\"53\",\"55\"],\"answer\":2,\"drill\":\"Drill 5\",\"hint\":\"The values are already ordered, so select the middle of the five measurements.\",\"explanation\":\"The ordered heights are 47, 51, 53, 61, and 63. The third and middle value is 53 inches, choice C.\"},{\"prompt\":\"Group 1 contains 18, 20, 24, 27, and 32. Group 2 contains 38, 44, 49, 55, and 63. How much less is the mean of group 1 than the mean of group 2?\",\"options\":[\"24.2\",\"25.6\",\"49.8\",\"128\"],\"answer\":1,\"drill\":\"Drill 5\",\"hint\":\"Find each group\\u0027s mean separately, then subtract the group 1 mean from the group 2 mean.\",\"explanation\":\"Group 1 has mean 121 \\u00f7 5 = 24.2, while group 2 has mean 249 \\u00f7 5 = 49.8. The difference is 49.8 \\u2212 24.2 = 25.6, choice B.\"},{\"prompt\":\"Data set 1 is 6, 7, 8, 9, 10, 11. Data set 2 is 1, 4, 8, 9, 12, 17. What must be true when comparing the first data set to the second?\",\"options\":[\"The standard deviations are the same, and the medians are different\",\"The standard deviations are different, and the medians are the same\",\"Both the standard deviations and the medians are the same\",\"Both the standard deviations and the medians are different\"],\"answer\":1,\"drill\":\"Drill 5\",\"hint\":\"Compare the two middle values in each ordered set, then compare how widely each set is spread.\",\"explanation\":\"Both sets have middle values 8 and 9, so both medians are 8.5. Data set 2 is more spread out than data set 1, so their standard deviations differ. Choice B is correct.\"},{\"prompt\":\"How do the standard deviations of the spring and fall data sets compare?\",\"options\":[\"The spring standard deviation is greater\",\"The standard deviations are equal\",\"The spring standard deviation is less\",\"There is not enough information\"],\"answer\":0,\"drill\":\"Drill 5\",\"hint\":\"Standard deviation measures spread around the center; compare how dispersed the dots are in each plot.\",\"explanation\":\"The spring observations are distributed more broadly across their scale, while the fall observations are more concentrated around the center. Therefore the spring data set has the greater standard deviation, choice A.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/data-analysis\\\/redrawn\\\/spring-fall-dotplots.svg\"},{\"prompt\":\"A solar monitoring station records an average of 18 kilowatt-hours per day over its first 4 days. Its average over all 9 days is 23 kilowatt-hours per day. What is the average production over the final 5 days, in kilowatt-hours per day?\",\"options\":[\"21\",\"23\",\"25\",\"27\"],\"answer\":3,\"drill\":\"Drill 5\",\"hint\":\"Subtract the first four-day total from the nine-day total.\",\"explanation\":\"The nine-day total is 9(23) = 207, and the first four-day total is 4(18) = 72. The final five-day total is 135, giving an average of 135\\\/5 = 27.\"},{\"prompt\":\"Eight test scores are 72, 76, 79, 81, 84, 87, 91, and 94. After a ninth makeup score is included, the mean of all nine scores is 81. What was the makeup score?\",\"options\":[\"65\",\"73\",\"81\",\"89\"],\"answer\":0,\"drill\":\"Drill 5\",\"hint\":\"Use the new mean to find the total of all nine scores, then subtract the sum of the eight displayed scores.\",\"explanation\":\"The eight displayed scores sum to 664. A mean of 81 for nine students requires a total of 9 \\u00d7 81 = 729. The makeup score was 729 \\u2212 664 = 65, choice A.\"},{\"prompt\":\"A data list has a minimum of 27. The value 19 is added to the list, and the maximum does not change. Which measure increases by 8?\",\"options\":[\"The standard deviation\",\"The range\",\"The median\",\"The mode\"],\"answer\":1,\"drill\":\"Drill 5\",\"hint\":\"The new value becomes the minimum; compare the old and new maximum-minus-minimum calculations.\",\"explanation\":\"The maximum stays unchanged, while the minimum decreases from 27 to 19, a decrease of 8. Therefore maximum \\u2212 minimum, the range, increases by 8. Choice B is correct.\"},{\"prompt\":\"What is the average annual temperature of the coldest state shown?\",\"options\":[\"20\\u00b0F\",\"30\\u00b0F\",\"40\\u00b0F\",\"50\\u00b0F\"],\"answer\":1,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/data-analysis\\\/redrawn\\\/state-temperatures.svg\"},{\"prompt\":\"At 6 megabytes per second, approximately how many minutes are needed to download sixteen 110-megabyte files?\",\"options\":[\"1\",\"5\",\"50\",\"300\"],\"answer\":1,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"A graph shows the current ages of a group of students. A second graph is made using the ages of the same students exactly 5 years ago. How do the means and ranges compare?\",\"options\":[\"Equal means; current range smaller\",\"Equal means; current range greater\",\"Current mean smaller; same range\",\"Current mean greater; same range\"],\"answer\":3,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"The orbital periods of eight planets, in Earth years, are 0.24, 0.62, 1.00, 1.88, 11.86, 29.46, 84.07, and 164.95. What is their average orbital period?\",\"options\":[\"29.48\",\"36.76\",\"42.01\",\"294.08\"],\"answer\":1,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"A data list has a minimum of 27. The value 19 is added to the list, and the maximum does not change. Which measure increases by 8?\",\"options\":[\"The standard deviation\",\"The range\",\"The median\",\"The mode\"],\"answer\":1,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"For the data set 4, 5, 5, 6, 30, which measure is affected most by the value 30?\",\"options\":[\"Mean\",\"Median\",\"Mode\",\"Minimum\"],\"answer\":0,\"drill\":\"Common Mistakes\",\"hint\":\"Consider which measure uses every value in its calculation.\",\"explanation\":\"The mean uses every value, so the unusually large value 30 pulls it upward. The median and mode remain 5.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A class of 12 students has an average score of 70. Another class of 8 students has an average score of 85. What is the combined average?\",\"options\":[\"75\",\"76\",\"77.5\",\"80\"],\"answer\":1,\"drill\":\"Challenge Question\",\"hint\":\"Find each class\\u0027s total points before combining.\",\"explanation\":\"The total is 12(70) + 8(85) = 840 + 680 = 1,520. Dividing by 20 students gives 76.\"}]},{\"title\":\"Trigonometry\",\"questions\":[{\"prompt\":\"What is the volume of a sphere with radius 6 centimeters?\",\"options\":[\"8\\u03c0\",\"48\\u03c0\",\"288\\u03c0\",\"2,304\\u03c0\"],\"answer\":2,\"drill\":\"Drill 1\",\"hint\":\"Use V = (4\\\/3)\\u03c0r\\u00b3 with r = 6.\",\"explanation\":\"V = (4\\\/3)\\u03c0(6\\u00b3) = (4\\\/3)\\u03c0(216) = 288\\u03c0 cubic centimeters, choice C.\"},{\"prompt\":\"Parallel lines \\u2113 and m are cut by a transversal. Two corresponding angles measure 127\\u00b0 and (x + 19)\\u00b0. What is the value of x?\",\"options\":[\"54\",\"108\",\"127\",\"146\"],\"answer\":1,\"drill\":\"Drill 1\",\"hint\":\"Corresponding angles formed by a transversal crossing parallel lines are congruent.\",\"explanation\":\"Corresponding angles are equal, so x + 19 = 127. Subtracting 19 from both sides gives x = 108, choice B.\"},{\"prompt\":\"A right-triangular roof support has acute angles A and B. Its dimensions give cos A = 28\\\/53. Without finding either angle in degrees, what is sin B?\",\"options\":[\"28\\\/45\",\"45\\\/53\",\"28\\\/53\",\"53\\\/28\"],\"answer\":2,\"drill\":\"Drill 1\",\"hint\":\"Use the relationship between the acute angles in a right triangle.\",\"explanation\":\"The acute angles are complementary, so sin B = cos A = 28\\\/53.\"},{\"prompt\":\"An isosceles right triangle has hypotenuse 16\\u221a2 meters. What is the length of one leg?\",\"options\":[\"8\",\"16\",\"32\",\"128\"],\"answer\":1,\"drill\":\"Drill 1\",\"hint\":\"In a 45\\u00b0-45\\u00b0-90\\u00b0 triangle, the hypotenuse equals a leg times \\u221a2.\",\"explanation\":\"If each leg has length s, the hypotenuse is s\\u221a2. Setting s\\u221a2 = 16\\u221a2 gives s = 16 meters, choice B.\"},{\"prompt\":\"What is the perimeter of a rectangle with length 2x feet and width x feet?\",\"options\":[\"2x\",\"3x\",\"6x\",\"2x\\u00b2\"],\"answer\":2,\"drill\":\"Drill 1\",\"hint\":\"Use P = 2(length + width).\",\"explanation\":\"P = 2(2x + x) = 2(3x) = 6x feet, choice C.\"},{\"prompt\":\"Triangle ABC is equilateral. Altitude AD meets side BC at D. If BD = 6, what is the length of AC?\",\"options\":[\"6\",\"6\\u221a2\",\"6\\u221a3\",\"12\"],\"answer\":3,\"drill\":\"Drill 2\",\"hint\":\"In an equilateral triangle, an altitude also bisects the opposite side.\",\"explanation\":\"Because ABC is equilateral, altitude AD bisects BC. Thus BC = 2(BD) = 12. All sides of an equilateral triangle are equal, so AC = 12, choice D.\"},{\"prompt\":\"A circular pool has area 1.21\\u03c0 square meters. What is its circumference?\",\"options\":[\"1.1\\u03c0\",\"1.21\\u03c0\",\"2.2\\u03c0\",\"2.42\\u03c0\"],\"answer\":2,\"drill\":\"Drill 2\",\"hint\":\"First use A = \\u03c0r\\u00b2 to find the radius, then use C = 2\\u03c0r.\",\"explanation\":\"From \\u03c0r\\u00b2 = 1.21\\u03c0, r\\u00b2 = 1.21 and r = 1.1 meters. The circumference is 2\\u03c0(1.1) = 2.2\\u03c0 meters, choice C.\"},{\"prompt\":\"Right triangles DEF and GHI are similar. What is sin(F)?\",\"options\":[\"3\\\/5\",\"3\\\/4\",\"4\\\/5\",\"5\\\/3\"],\"answer\":2,\"drill\":\"Drill 2\",\"hint\":\"Use the corresponding angle I and identify its opposite side and hypotenuse in triangle GHI.\",\"explanation\":\"Because D corresponds to G and E corresponds to H, angle F corresponds to angle I. In triangle GHI, the side opposite I is GH = 76 and the hypotenuse is HI = 95. Thus sin(F) = sin(I) = 76\\\/95 = 4\\\/5, choice C.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/trigonometry\\\/redrawn\\\/similar-right-triangles.svg\"},{\"prompt\":\"A rectangular pyramid has a 5-inch by 5-inch base, height x, and volume 50 cubic inches. What is x?\",\"options\":[\"2\",\"5\",\"6\",\"10\"],\"answer\":2,\"drill\":\"Drill 2\",\"hint\":\"Use V = (1\\\/3)Bh, where B is the area of the rectangular base.\",\"explanation\":\"The base area is 5\\u00b75 = 25 square inches. Using V = (1\\\/3)Bh gives 50 = (1\\\/3)(25)x. Multiplying by 3 gives 150 = 25x, so x = 6.\"},{\"prompt\":\"Four identical rectangular boxes are stacked directly on top of one another. Each box is 7 inches high, and the total volume of the stack is 3,024 cubic inches. What is the area, in square inches, of the base of one box?\",\"options\":[\"27\",\"108\",\"432\",\"756\"],\"answer\":1,\"drill\":\"Drill 2\",\"hint\":\"The total height of the stack is 4 \\u00b7 7 inches. Use volume = base area \\u00b7 height.\",\"explanation\":\"The four boxes have a combined height of 4(7) = 28 inches. If B is the base area, then 3,024 = 28B. Therefore B = 3,024\\\/28 = 108 square inches, choice B.\"},{\"prompt\":\"A triangle has height 12 inches and base 9 inches. What is its area?\",\"options\":[\"19 in\\u00b2\",\"27 in\\u00b2\",\"36 in\\u00b2\",\"54 in\\u00b2\"],\"answer\":3,\"drill\":\"Drill 3\",\"hint\":\"Use A = (1\\\/2)bh.\",\"explanation\":\"A = (1\\\/2)(9)(12) = 54 square inches, choice D.\"},{\"prompt\":\"Parallel lines k and \\u2113 are cut by transversal p. Angle a and a 108\\u00b0 angle are alternate interior angles. What is a?\",\"options\":[\"72\",\"92\",\"108\",\"252\"],\"answer\":2,\"drill\":\"Drill 3\",\"hint\":\"The marked angles lie between the parallel lines on opposite sides of transversal p.\",\"explanation\":\"Angle a and the 108\\u00b0 angle are alternate interior angles formed by transversal p crossing parallel lines k and \\u2113. Alternate interior angles are congruent, so a = 108, choice C.\"},{\"prompt\":\"Parallel lines \\u2113 and m are cut by a transversal. Angle a corresponds to an angle that forms a linear pair with angle b. What is a + b?\",\"options\":[\"90\",\"180\",\"270\",\"360\"],\"answer\":1,\"drill\":\"Drill 3\",\"hint\":\"Transfer angle a to the lower intersection using corresponding angles, then use a linear pair.\",\"explanation\":\"The angle above line \\u2113 and to the left of the transversal corresponds to angle a, so it also measures a\\u00b0. That angle and b form a linear pair on line \\u2113, so a + b = 180\\u00b0, choice B.\"},{\"prompt\":\"Parallel lines a and b are cut by a transversal. Angles f and g form a linear pair, and angle h corresponds to angle f. What is h if f = 9t \\u2212 5 and g = 3t + 5?\",\"options\":[\"15\",\"50\",\"130\",\"165\"],\"answer\":2,\"drill\":\"Drill 3\",\"hint\":\"Angles f and g form a linear pair; then relate f and h using corresponding angles.\",\"explanation\":\"Since f and g form a linear pair, (9t \\u2212 5) + (3t + 5) = 180. Thus 12t = 180 and t = 15. Then f = 9(15) \\u2212 5 = 130\\u00b0. Angles f and h are corresponding angles on parallel lines, so h = 130\\u00b0, choice C.\"},{\"prompt\":\"In triangles ABC and XYZ, angles B and Y are both 35\\u00b0. Which additional information is sufficient to prove the triangles are similar?\",\"options\":[\"XY = 4\",\"Angles A and C are 100\\u00b0 and 25\\u00b0\",\"Angles A and Z are 100\\u00b0 and 45\\u00b0\",\"No additional information is necessary\"],\"answer\":2,\"drill\":\"Drill 3\",\"hint\":\"One angle pair is already equal. Look for information that produces a second equal angle pair.\",\"explanation\":\"Angles B and Y are both 35\\u00b0. If angle A = 100\\u00b0, then angle C = 180\\u00b0 \\u2212 100\\u00b0 \\u2212 35\\u00b0 = 45\\u00b0. Choice C also states angle Z = 45\\u00b0, giving C = Z. Two pairs of corresponding angles are equal, so the triangles are similar by AA.\"},{\"prompt\":\"A square has perimeter 64. What is the length of one side?\",\"options\":[\"8\",\"16\",\"32\",\"64\"],\"answer\":1,\"drill\":\"Drill 4\",\"hint\":\"A square has four equal sides.\",\"explanation\":\"If s is the side length, 4s = 64. Dividing by 4 gives s = 16.\"},{\"prompt\":\"A cylinder has radius 3 inches and volume 81\\u03c0 cubic inches. What is its height?\",\"options\":[\"3\",\"9\",\"27\",\"81\"],\"answer\":1,\"drill\":\"Drill 4\",\"hint\":\"Use V = \\u03c0r\\u00b2h.\",\"explanation\":\"81\\u03c0 = \\u03c0(3\\u00b2)h = 9\\u03c0h. Dividing by 9\\u03c0 gives h = 9 inches, choice B.\"},{\"prompt\":\"Square B has a side length three times that of square A. If square A has area 36, what is one side of square B?\",\"options\":[\"10.4\",\"12\",\"18\",\"27\"],\"answer\":2,\"drill\":\"Drill 4\",\"hint\":\"Find the side length of square A by taking the square root of its area.\",\"explanation\":\"Square A has side length \\u221a36 = 6 inches. Square B\\u0027s side is 3(6) = 18 inches, choice C.\"},{\"prompt\":\"A cube has surface area 6x square inches. What is its volume?\",\"options\":[\"x^(2\\\/3)\",\"x^(3\\\/2)\",\"x\\u00b3\",\"6x\\u00b3\"],\"answer\":1,\"drill\":\"Drill 4\",\"hint\":\"Set 6s\\u00b2 = 6x to find the cube\\u0027s side length s.\",\"explanation\":\"For side length s, the surface area is 6s\\u00b2. Thus 6s\\u00b2 = 6x, so s\\u00b2 = x and s = \\u221ax. The volume is s\\u00b3 = (\\u221ax)\\u00b3 = x^(3\\\/2), choice B.\"},{\"prompt\":\"A square has a diagonal of 8 centimeters. What is its area?\",\"options\":[\"4\\u221a2\",\"8\",\"8\\u221a2\",\"32\"],\"answer\":3,\"drill\":\"Drill 4\",\"hint\":\"The diagonal divides the square into two congruent right triangles, so d\\u00b2 = 2s\\u00b2.\",\"explanation\":\"With diagonal d = 8, d\\u00b2 = 2s\\u00b2 gives 64 = 2s\\u00b2. Therefore s\\u00b2 = 32, which is the square\\u0027s area, choice D.\"},{\"prompt\":\"Rectangle Y has dimensions one-ninth those of a 270-by-378-foot rectangle. What is the area of rectangle Y?\",\"options\":[\"140\",\"420\",\"1,260\",\"11,340\"],\"answer\":2,\"drill\":\"Drill 4\",\"hint\":\"Divide each original dimension by 9 before multiplying.\",\"explanation\":\"Rectangle Y has width 270\\\/9 = 30 feet and length 378\\\/9 = 42 feet. Its area is 30\\u00b742 = 1,260 square feet.\"},{\"prompt\":\"A gardener plants 5 bulbs per square foot in a 4-by-10-foot plot. How many bulbs are needed?\",\"options\":[\"40\",\"70\",\"120\",\"200\"],\"answer\":3,\"drill\":\"Drill 4\",\"hint\":\"Find the plot\\u0027s area, then multiply by the planting rate.\",\"explanation\":\"The plot\\u0027s area is 4\\u00b710 = 40 square feet. At 5 bulbs per square foot, the gardener needs 40\\u00b75 = 200 bulbs, choice D.\"},{\"prompt\":\"Two 3-by-4-by-5 solids are joined to form a 3-by-4-by-10 rectangular solid. What is its surface area?\",\"options\":[\"82\",\"94\",\"120\",\"164\"],\"answer\":3,\"drill\":\"Drill 4\",\"hint\":\"Use SA = 2(lw + lh + wh) with dimensions 3, 4, and 10.\",\"explanation\":\"The combined solid has dimensions 3, 4, and 10. Its surface area is 2(3\\u00b74 + 3\\u00b710 + 4\\u00b710) = 2(12 + 30 + 40) = 164 square units, choice D.\"},{\"prompt\":\"In right triangle XYZ, angle Y is 23\\u00b0 and angle Z is 90\\u00b0. What is angle X?\",\"options\":[\"67\\u00b0\",\"90\\u00b0\",\"113\\u00b0\",\"157\\u00b0\"],\"answer\":0,\"drill\":\"Drill 5\",\"hint\":\"The three interior angles of a triangle sum to 180\\u00b0.\",\"explanation\":\"X + 23\\u00b0 + 90\\u00b0 = 180\\u00b0, so X = 180\\u00b0 \\u2212 113\\u00b0 = 67\\u00b0, choice A.\"},{\"prompt\":\"Which expression represents t if u = 15 and v = 17 in the right triangle?\",\"options\":[\"\\u221a(17\\u00b2 \\u2212 15\\u00b2)\",\"\\u221a(17\\u00b2 + 15\\u00b2)\",\"\\u221a(17 \\u2212 15)\",\"17 \\u2212 15\"],\"answer\":0,\"drill\":\"Drill 5\",\"hint\":\"The side v is opposite the right angle, so it is the hypotenuse.\",\"explanation\":\"By the Pythagorean theorem, t\\u00b2 + u\\u00b2 = v\\u00b2. Therefore t = \\u221a(v\\u00b2 \\u2212 u\\u00b2) = \\u221a(17\\u00b2 \\u2212 15\\u00b2), choice A. This also simplifies to \\u221a64 = 8.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/trigonometry\\\/redrawn\\\/right-triangle-15-17.svg\"},{\"prompt\":\"In triangle DEF, angle D is 90\\u00b0 and cos(F) = 5\\\/8. What is cos(E)?\",\"options\":[\"\\u221a39\\\/39\",\"5\\\/8\",\"\\u221a39\\\/8\",\"5\\u221a39\\\/39\"],\"answer\":2,\"drill\":\"Drill 5\",\"hint\":\"Use an 8-unit hypotenuse and 5-unit leg, then find the other leg with the Pythagorean theorem.\",\"explanation\":\"Let the hypotenuse EF be 8 and the leg adjacent to F, DF, be 5. Then DE = \\u221a(8\\u00b2 \\u2212 5\\u00b2) = \\u221a39. For angle E, the adjacent leg is DE, so cos(E) = \\u221a39\\\/8, choice C.\"},{\"prompt\":\"Right triangles ABC and DEF are similar, with angle A corresponding to angle D. In triangle ABC, angle B is 90\\u00b0, AB = 21, BC = 72, and AC = 75. What is cos(D)?\",\"options\":[\"7\\\/25\",\"7\\\/24\",\"24\\\/25\",\"25\\\/7\"],\"answer\":0,\"drill\":\"Drill 5\",\"hint\":\"Since D corresponds to A, find cos(A) using the adjacent leg and the hypotenuse of triangle ABC.\",\"explanation\":\"Because angle B is 90\\u00b0, AC = 75 is the hypotenuse. For angle A, the adjacent leg is AB = 21, so cos(A) = 21\\\/75 = 7\\\/25. Since A corresponds to D, cos(D) = 7\\\/25, choice A.\"},{\"prompt\":\"A rectangle has perimeter 52, and its length \\u2113 is 2 less than 6 times its width w. What is \\u2113?\",\"options\":[\"4\",\"9\",\"18\",\"22\"],\"answer\":3,\"drill\":\"Drill 6\",\"hint\":\"Use \\u2113 + w = 26 together with \\u2113 = 6w \\u2212 2.\",\"explanation\":\"From 2\\u2113 + 2w = 52, \\u2113 + w = 26. Substitute \\u2113 = 6w \\u2212 2: 6w \\u2212 2 + w = 26, so 7w = 28 and w = 4. Therefore \\u2113 = 6(4) \\u2212 2 = 22, choice D.\"},{\"prompt\":\"In right triangle ABC, sin(B) = 70\\\/74 and angle C is 90\\u00b0. What is sin(A)?\",\"options\":[\"12\\\/37\",\"24\\\/37\",\"35\\\/37\",\"37\\\/12\"],\"answer\":0,\"drill\":\"Drill 6\",\"hint\":\"Treat 70 and 74 as the opposite leg and hypotenuse, then find the remaining leg.\",\"explanation\":\"For angle B, the opposite leg is AC = 70 and the hypotenuse is AB = 74. The remaining leg is BC = \\u221a(74\\u00b2 \\u2212 70\\u00b2) = \\u221a576 = 24. Therefore sin(A) = BC\\\/AB = 24\\\/74 = 12\\\/37.\"},{\"prompt\":\"Angle A is 7\\u03c0\\\/4 radians, and angle B is 5\\u03c0\\\/6 radians less. What is angle B in degrees?\",\"options\":[\"150\",\"165\",\"315\",\"330\"],\"answer\":1,\"drill\":\"Drill 6\",\"hint\":\"Subtract the radian measures using a common denominator, then multiply by 180\\\/\\u03c0.\",\"explanation\":\"B = 7\\u03c0\\\/4 \\u2212 5\\u03c0\\\/6 = 21\\u03c0\\\/12 \\u2212 10\\u03c0\\\/12 = 11\\u03c0\\\/12. Converting to degrees gives (11\\u03c0\\\/12)(180\\\/\\u03c0) = 165\\u00b0, choice B.\"},{\"prompt\":\"In circle O, radii OR and OS form an 88\\u00b0 central angle. What is angle RSO?\",\"options\":[\"44\\u00b0\",\"46\\u00b0\",\"88\\u00b0\",\"92\\u00b0\"],\"answer\":1,\"drill\":\"Drill 6\",\"hint\":\"Triangle ORS is isosceles because OR and OS are radii.\",\"explanation\":\"Since OR = OS, the base angles at R and S are equal. They share the remaining 180\\u00b0 \\u2212 88\\u00b0 = 92\\u00b0, so each is 92\\u00b0\\\/2 = 46\\u00b0. Thus angle RSO = 46\\u00b0.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/drills\\\/trigonometry\\\/redrawn\\\/central-angle-circle.svg\"},{\"prompt\":\"Right triangles ABC and DEF are similar, with A corresponding to D. In right triangle ABC, BC = 36, CA = 45, and angle B is 90\\u00b0. What is sin(D)?\",\"options\":[\"3\\\/5\",\"3\\\/4\",\"4\\\/5\",\"4\\\/3\"],\"answer\":2,\"drill\":\"Drill 6\",\"hint\":\"Find sin(A) in the 27-36-45 triangle.\",\"explanation\":\"The hypotenuse is CA = 45, and the side opposite angle A is BC = 36. Since D corresponds to A, sin(D) = sin(A) = 36\\\/45 = 4\\\/5, choice C.\"},{\"prompt\":\"What is the volume of a sphere with radius 6 centimeters?\",\"options\":[\"8\\u03c0\",\"48\\u03c0\",\"288\\u03c0\",\"2,304\\u03c0\"],\"answer\":2,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"A rectangular pyramid has a 5-inch by 5-inch base, height x, and volume 50 cubic inches. What is x?\",\"options\":[\"2\",\"5\",\"6\",\"10\"],\"answer\":2,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"A cylinder has radius 3 inches and volume 81\\u03c0 cubic inches. What is its height?\",\"options\":[\"3\",\"9\",\"27\",\"81\"],\"answer\":1,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"In right triangle XYZ, angle Y is 23\\u00b0 and angle Z is 90\\u00b0. What is angle X?\",\"options\":[\"67\\u00b0\",\"90\\u00b0\",\"113\\u00b0\",\"157\\u00b0\"],\"answer\":0,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"prompt\":\"Right triangles ABC and DEF are similar, with A corresponding to D. In right triangle ABC, BC = 36, CA = 45, and angle B is 90\\u00b0. What is sin(D)?\",\"options\":[\"3\\\/5\",\"3\\\/4\",\"4\\\/5\",\"4\\\/3\"],\"answer\":2,\"drill\":\"Exit Ticket\",\"hint\":\"\",\"explanation\":\"\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"In a right triangle, relative to angle \\u03b8, the opposite side is 9 and the hypotenuse is 15. What is sin \\u03b8?\",\"options\":[\"3\\\/5\",\"4\\\/5\",\"5\\\/3\",\"5\\\/4\"],\"answer\":0,\"drill\":\"Common Mistakes\",\"hint\":\"Sine equals opposite divided by hypotenuse.\",\"explanation\":\"sin \\u03b8 = opposite\\\/hypotenuse = 9\\\/15 = 3\\\/5. A common mistake is to reverse the ratio.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"In a right triangle, tan \\u03b8 = 3\\\/4 and the hypotenuse is 20. What is the length of the side opposite \\u03b8?\",\"options\":[\"8\",\"12\",\"15\",\"16\"],\"answer\":1,\"drill\":\"Challenge Question\",\"hint\":\"Use the 3-4-5 triangle ratio.\",\"explanation\":\"The side ratio is 3:4:5. Since the hypotenuse 20 equals 5 \\u00d7 4, the opposite side is 3 \\u00d7 4 = 12.\"}]},{\"title\":\"SAT Assessments\",\"questions\":[{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"An automobile factory produces 1,200 vehicles per day. If it operates continuously at this rate for 1 week, how many vehicles will it produce?\",\"options\":[\"7,200\",\"8,400\",\"9,600\",\"12,000\"],\"answer\":1,\"drill\":\"Test 1\",\"hint\":\"A week has 7 days.\",\"explanation\":\"At 1,200 vehicles per day for 7 days, the factory produces 1,200 \\u00d7 7 = 8,400 vehicles.\"},{\"domain\":\"Algebra\",\"prompt\":\"If 10a = 30, what is the value of 9a?\",\"options\":[\"27\",\"40\",\"270\",\"360\"],\"answer\":0,\"drill\":\"Test 1\",\"hint\":\"First solve for a.\",\"explanation\":\"Dividing 10a = 30 by 10 gives a = 3. Therefore, 9a = 27.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"Talula baked 340 cookies and gave 20% of them to her neighbors. How many cookies did she give to her neighbors?\",\"options\":[\"14\",\"34\",\"54\",\"68\"],\"answer\":3,\"drill\":\"Test 1\",\"hint\":\"Find 20% of 340.\",\"explanation\":\"20% of 340 is 0.20 \\u00d7 340 = 68 cookies.\"},{\"domain\":\"Algebra\",\"prompt\":\"The function h is defined by h(x) = 11x \\u2212 5. What is the value of h(6)?\",\"options\":[\"55\",\"61\",\"66\",\"71\"],\"answer\":1,\"drill\":\"Test 1\",\"hint\":\"Substitute 6 for x.\",\"explanation\":\"h(6) = 11(6) \\u2212 5 = 66 \\u2212 5 = 61.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The bar graph shows the number of dogs that visited a dog park during an 11-day period. How many dogs visited the park on day 7?\",\"options\":[\"35\",\"40\",\"45\",\"55\"],\"answer\":2,\"drill\":\"Test 1\",\"hint\":\"Read the height of the bar above day 7.\",\"explanation\":\"The bar for day 7 reaches 45, so 45 dogs visited the park.\",\"visual\":\"test1-dog-bars\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The table shows the number of books read by students during a month. What is the mean number of books read per student?\",\"options\":[\"3.2\",\"3.5\",\"3.8\",\"4.0\"],\"answer\":1,\"drill\":\"Test 1\",\"hint\":\"Multiply each number of books by its frequency, add the products, and divide by the total frequency.\",\"explanation\":\"The total number of books is 1(2) + 3(4) + 5(3) + 6(1) = 35. There are 2 + 4 + 3 + 1 = 10 students, so the mean is 35\\\/10 = 3.5.\",\"visual\":\"test1-mean-table\"},{\"domain\":\"Algebra\",\"prompt\":\"The graph of line \\u2113 shows its x-intercept and y-intercept. Which equation represents line \\u2113?\",\"options\":[\"y = 2x + 6\",\"y = \\u22122x + 6\",\"y = \\u2212x\\\/2 + 6\",\"y = \\u22122x + 3\"],\"answer\":1,\"drill\":\"Test 1\",\"hint\":\"Use the labeled intercepts (0, 6) and (3, 0) to calculate the slope.\",\"explanation\":\"The slope is (0 \\u2212 6)\\\/(3 \\u2212 0) = \\u22122, and the y-intercept is 6. Therefore, the equation is y = \\u22122x + 6.\",\"visual\":\"test1-intercept-line\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"In right triangle ABC, angle C is a right angle. If cos A = 5\\\/13, what is sin B?\",\"options\":[\"5\\\/13\",\"12\\\/13\",\"5\\\/12\",\"13\\\/5\"],\"answer\":0,\"drill\":\"Test 1\",\"hint\":\"The two acute angles of a right triangle are complementary.\",\"explanation\":\"Angles A and B are complementary, so sin B = cos A. Since cos A = 5\\\/13, sin B = 5\\\/13.\",\"visual\":\"test1-complementary-triangle\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The two data sets are shown. How does the range of data set B compare with the range of data set A?\",\"options\":[\"It is 3 less.\",\"It is the same.\",\"It is 3 greater.\",\"It is 7 greater.\"],\"answer\":2,\"drill\":\"Test 1\",\"hint\":\"For each data set, subtract the least value from the greatest value.\",\"explanation\":\"Data set A has range 16 \\u2212 4 = 12. Data set B has range 18 \\u2212 3 = 15. Therefore, the range of data set B is 3 greater.\",\"visual\":\"test1-range-sets\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A hat contains 18 cards numbered 1 through 18, with a different number on each card. If one card is selected at random, what is the probability that its number is not 6?\",\"options\":[\"1\\\/18\",\"6\\\/18\",\"12\\\/18\",\"17\\\/18\"],\"answer\":3,\"drill\":\"Test 1\",\"hint\":\"Only one of the 18 cards is numbered 6.\",\"explanation\":\"Seventeen of the 18 cards are not numbered 6, so the probability is 17\\\/18.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The function g is defined by g(x) = \\u22122x\\u00b2. What is the value of g(3)?\",\"options\":[\"\\u221218\",\"\\u221212\",\"\\u221210\",\"\\u22126\"],\"answer\":0,\"drill\":\"Test 1\",\"hint\":\"Substitute 3 for x before multiplying by \\u22122.\",\"explanation\":\"g(3) = \\u22122(3\\u00b2) = \\u22122(9) = \\u221218.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"Line b intersects parallel lines p and r. One obtuse angle measures 148\\u00b0. What is the value of the acute angle d?\",\"options\":[\"16\",\"32\",\"74\",\"148\"],\"answer\":1,\"drill\":\"Test 1\",\"hint\":\"An acute and adjacent obtuse angle form a linear pair.\",\"explanation\":\"The angles are supplementary, so d = 180\\u00b0 \\u2212 148\\u00b0 = 32\\u00b0.\",\"visual\":\"test1-parallel\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"Which equation could represent the line of best fit for the scatterplot shown?\",\"options\":[\"y = \\u221213 \\u2212 0.7x\",\"y = \\u221213 + 0.7x\",\"y = 13 \\u2212 0.7x\",\"y = 13 + 0.7x\"],\"answer\":2,\"drill\":\"Test 1\",\"hint\":\"Estimate the y-intercept and determine whether the trend is increasing or decreasing.\",\"explanation\":\"The line has a positive y-intercept near 13 and decreases by about 0.7 for each unit increase in x, so y = 13 \\u2212 0.7x.\",\"visual\":\"test1-best-fit\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The table shows the number of hours x that five students studied and their test scores y. Which equation is the best linear regression model for the data?\",\"options\":[\"y = 2.8x + 3.2\",\"y = 3.2x + 2.8\",\"y = \\u22122.8x + 19.6\",\"y = 5x + 1\"],\"answer\":0,\"drill\":\"Test 1\",\"hint\":\"A regression line should follow the positive trend and pass near the center of the data.\",\"explanation\":\"For the five data points, the least-squares slope is 2.8 and the y-intercept is 3.2. Therefore, the best linear regression model is y = 2.8x + 3.2.\",\"visual\":\"test1-regression-table\"},{\"domain\":\"Algebra\",\"prompt\":\"Line m is represented by 4x \\u2212 3y = 12. What is the slope of any line perpendicular to line m?\",\"options\":[\"\\u22124\\\/3\",\"\\u22123\\\/4\",\"3\\\/4\",\"4\\\/3\"],\"answer\":1,\"drill\":\"Test 1\",\"hint\":\"Rewrite the equation in slope-intercept form, then take the negative reciprocal of the slope.\",\"explanation\":\"Rewriting gives y = (4\\\/3)x \\u2212 4, so line m has slope 4\\\/3. A perpendicular line has slope \\u22123\\\/4.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"What is the median of the data set 14, 8, 21, 17, 12, 10, 19?\",\"options\":[\"12\",\"14\",\"15\",\"17\"],\"answer\":1,\"drill\":\"Test 1\",\"hint\":\"Order the seven values from least to greatest and select the middle value.\",\"explanation\":\"In order, the values are 8, 10, 12, 14, 17, 19, 21. The fourth and middle value is 14, so the median is 14.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The graphs of y = x\\u00b2 \\u2212 7 and x = \\u22127 intersect at (x, y). What is the value of y?\",\"options\":[\"\\u221221\",\"\\u22127\",\"42\",\"49\"],\"answer\":2,\"drill\":\"Test 1\",\"hint\":\"Substitute x = \\u22127 into y = x\\u00b2 \\u2212 7.\",\"explanation\":\"y = (\\u22127)\\u00b2 \\u2212 7 = 49 \\u2212 7 = 42.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"In the figure, lines p and q are parallel and are intersected by transversal t. What is the value of x?\",\"options\":[\"58\",\"62\",\"118\",\"122\"],\"answer\":1,\"drill\":\"Test 1\",\"hint\":\"The two labeled angles are same-side interior angles formed by parallel lines.\",\"explanation\":\"Same-side interior angles are supplementary, so x + 118 = 180. Therefore, x = 62.\",\"visual\":\"test1-interior-angles\"},{\"domain\":\"Algebra\",\"prompt\":\"Which equation defines a line with slope 1\\\/6 that passes through the point (12, \\u22127)?\",\"options\":[\"y = x\\\/6 \\u2212 9\",\"y = x\\\/6 \\u2212 7\",\"y = \\u22129x + 1\\\/6\",\"y = 12x \\u2212 7\"],\"answer\":0,\"drill\":\"Test 1\",\"hint\":\"Substitute (12, \\u22127) into y = x\\\/6 + b.\",\"explanation\":\"\\u22127 = 12\\\/6 + b = 2 + b, so b = \\u22129. The equation is y = x\\\/6 \\u2212 9.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"How many distinct real solutions does 4x\\u00b2 \\u2212 8x \\u2212 5 = 0 have?\",\"options\":[\"Exactly one\",\"Exactly two\",\"Infinitely many\",\"Zero\"],\"answer\":1,\"drill\":\"Test 1\",\"hint\":\"Use the sign of the discriminant.\",\"explanation\":\"The discriminant is (\\u22128)\\u00b2 \\u2212 4(4)(\\u22125) = 144, which is positive. Therefore, there are exactly two distinct real solutions.\"},{\"domain\":\"Algebra\",\"prompt\":\"A jar contains 37 red and blue tokens. Each red token has mass 90 grams, each blue token has mass 120 grams, and the total mass is 3,810 grams. How many blue tokens are in the jar?\",\"options\":[\"5\",\"16\",\"21\",\"32\"],\"answer\":1,\"drill\":\"Test 1\",\"hint\":\"Let r + b = 37 and 90r + 120b = 3,810.\",\"explanation\":\"Using r = 37 \\u2212 b gives 90(37 \\u2212 b) + 120b = 3,810. Thus 3,330 + 30b = 3,810, so b = 16.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"Circle R is defined by (x + 3)\\u00b2 + y\\u00b2 = 64. Circle S is produced by shifting circle R 7 units to the right. Which equation defines circle S?\",\"options\":[\"(x + 3)\\u00b2 + (y \\u2212 7)\\u00b2 = 64\",\"(x + 3)\\u00b2 + (y + 7)\\u00b2 = 64\",\"(x \\u2212 4)\\u00b2 + y\\u00b2 = 64\",\"(x + 10)\\u00b2 + y\\u00b2 = 64\"],\"answer\":2,\"drill\":\"Test 1\",\"hint\":\"Find the original center and add 7 to its x-coordinate.\",\"explanation\":\"Circle R has center (\\u22123, 0). Shifting right 7 units gives center (4, 0), so circle S is (x \\u2212 4)\\u00b2 + y\\u00b2 = 64.\"},{\"domain\":\"Algebra\",\"topic\":\"Plugging In\",\"difficulty\":\"Easy\",\"wordProblem\":true,\"prompt\":\"A bicycle rental shop charges a fixed fee of 14 AED plus 9 AED per hour. The cost for h hours is C = 14 + 9h. What is the cost, in AED, of renting a bicycle for 4 hours?\",\"options\":[\"36\",\"50\",\"56\",\"92\"],\"answer\":1,\"drill\":\"Test 2\",\"hint\":\"Substitute 4 for h.\",\"explanation\":\"C = 14 + 9(4) = 50 AED.\"},{\"domain\":\"Algebra\",\"topic\":\"Plugging In Answers\",\"difficulty\":\"Easy\",\"wordProblem\":true,\"prompt\":\"A science club pays a 48 AED booking fee plus 12 AED per student for a workshop. The total is 216 AED. How many students attend?\",\"options\":[\"12\",\"13\",\"14\",\"18\"],\"answer\":2,\"drill\":\"Test 2\",\"hint\":\"Test the choices in 48 + 12n = 216.\",\"explanation\":\"For n = 14, the total is 48 + 12(14) = 216 AED. Therefore 14 students attend.\"},{\"domain\":\"Algebra\",\"topic\":\"Functions\",\"difficulty\":\"Easy\",\"wordProblem\":true,\"prompt\":\"A tank initially holds 180 liters of water. The function V(t) = 180 \\u2212 6t gives the volume remaining after t minutes while water drains at a constant rate. What does 6 represent?\",\"options\":[\"The initial volume, in liters\",\"The time until the tank is empty, in minutes\",\"The volume remaining after 1 minute, in liters\",\"The amount of water drained per minute, in liters\"],\"answer\":3,\"drill\":\"Test 2\",\"hint\":\"Interpret the rate of change.\",\"explanation\":\"The volume decreases by 6 liters each minute, so 6 is the amount drained per minute.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"topic\":\"Percentages \\u0026 Ratios\",\"difficulty\":\"Easy\",\"wordProblem\":true,\"prompt\":\"A backpack originally costs 240 AED. It is discounted by 15%. What is its sale price, in AED?\",\"options\":[\"204\",\"216\",\"225\",\"276\"],\"answer\":0,\"drill\":\"Test 2\",\"hint\":\"A 15% discount leaves 85% of the original price.\",\"explanation\":\"240(0.85) = 204 AED.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"topic\":\"Data Analysis\",\"difficulty\":\"Easy\",\"wordProblem\":true,\"prompt\":\"Seven students read 12, 18, 10, 22, 18, 16, and 30 pages. What is the median number of pages read?\",\"options\":[\"16\",\"17\",\"18\",\"20\"],\"answer\":2,\"drill\":\"Test 2\",\"hint\":\"Order the seven values and select the middle one.\",\"explanation\":\"The ordered values are 10, 12, 16, 18, 18, 22, 30. The fourth value, 18, is the median.\"},{\"domain\":\"Geometry and Trigonometry\",\"topic\":\"Trigonometry\",\"difficulty\":\"Easy\",\"wordProblem\":false,\"prompt\":\"In a right triangle, A and B are the two acute angles. If cos A = 5\\\/13, what is sin B?\",\"options\":[\"5\\\/13\",\"12\\\/13\",\"5\\\/12\",\"13\\\/5\"],\"answer\":0,\"drill\":\"Test 2\",\"hint\":\"The acute angles are complementary.\",\"explanation\":\"Because A + B = 90\\u00b0, sin B = cos A = 5\\\/13.\"},{\"domain\":\"Algebra\",\"topic\":\"Plugging In\",\"difficulty\":\"Average\",\"wordProblem\":true,\"prompt\":\"A square patio has side length s meters. Each side is increased by 3 meters. Which expression gives the increase in area, in square meters?\",\"options\":[\"3s + 9\",\"6s + 9\",\"s\\u00b2 + 9\",\"6s + 6\"],\"answer\":1,\"drill\":\"Test 2\",\"hint\":\"Try s = 2 in the choices, or subtract the original area from the new area.\",\"explanation\":\"The increase is (s + 3)\\u00b2 \\u2212 s\\u00b2 = 6s + 9. For example, s = 2 gives 25 \\u2212 4 = 21, which agrees with 6s + 9.\"},{\"domain\":\"Advanced Math\",\"topic\":\"Plugging In Answers\",\"difficulty\":\"Average\",\"wordProblem\":true,\"prompt\":\"A rectangular garden is 5 meters longer than it is wide and has an area of 84 square meters. What is its width, in meters?\",\"options\":[\"6\",\"7\",\"9\",\"12\"],\"answer\":1,\"drill\":\"Test 2\",\"hint\":\"Test each width w in w(w + 5) = 84.\",\"explanation\":\"A width of 7 gives a length of 12 and an area of 7(12) = 84 square meters.\"},{\"domain\":\"Algebra\",\"topic\":\"Linear Equations\",\"difficulty\":\"Average\",\"wordProblem\":true,\"prompt\":\"A school sells 120 tickets for a performance. Student tickets cost 8 AED and adult tickets cost 15 AED. Total ticket revenue is 1,240 AED. How many adult tickets were sold?\",\"options\":[\"35\",\"45\",\"80\",\"40\"],\"answer\":3,\"drill\":\"Test 2\",\"hint\":\"Let a be the number of adult tickets; the number of student tickets is 120 \\u2212 a.\",\"explanation\":\"15a + 8(120 \\u2212 a) = 1,240 gives 7a = 280, so a = 40.\"},{\"domain\":\"Algebra\",\"topic\":\"Linear Equations\",\"difficulty\":\"Average\",\"wordProblem\":true,\"prompt\":\"A club has at most 950 AED to spend on a trip. The bus costs 230 AED and admission costs 24 AED per student. What is the greatest number of students the club can pay for?\",\"options\":[\"29\",\"30\",\"31\",\"39\"],\"answer\":1,\"drill\":\"Test 2\",\"hint\":\"Use 230 + 24n \\u2264 950.\",\"explanation\":\"After paying for the bus, 720 AED remains. Since 720\\\/24 = 30, the club can pay for at most 30 students.\"},{\"domain\":\"Advanced Math\",\"topic\":\"Nonlinear Equations\",\"difficulty\":\"Average\",\"wordProblem\":true,\"prompt\":\"A laboratory sample initially contains 640 milligrams of a substance. The amount decreases by half every 3 hours. How many milligrams remain after 12 hours?\",\"options\":[\"20\",\"80\",\"160\",\"40\"],\"answer\":3,\"drill\":\"Test 2\",\"hint\":\"Count the number of 3-hour periods.\",\"explanation\":\"There are 12\\\/3 = 4 half-life periods. The remaining amount is 640(1\\\/2)^4 = 40 milligrams.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"topic\":\"Percentages \\u0026 Ratios\",\"difficulty\":\"Average\",\"wordProblem\":true,\"prompt\":\"A paint mixture contains blue and white paint in a ratio of 3 to 5. A painter needs 32 liters of the same mixture. How many liters of white paint are needed?\",\"options\":[\"12\",\"20\",\"24\",\"28\"],\"answer\":1,\"drill\":\"Test 2\",\"hint\":\"White paint makes up 5 of the 8 equal parts.\",\"explanation\":\"The white portion is 5\\\/8 of the mixture, so (5\\\/8)(32) = 20 liters.\"},{\"domain\":\"Geometry and Trigonometry\",\"topic\":\"Coordinate Geometry\",\"difficulty\":\"Average\",\"wordProblem\":true,\"prompt\":\"On a coordinate map, a straight road is represented by 4x + 3y = 18. A second road is perpendicular to it. What is the slope of the second road?\",\"options\":[\"\\u22124\\\/3\",\"\\u22123\\\/4\",\"3\\\/4\",\"4\\\/3\"],\"answer\":2,\"drill\":\"Test 2\",\"hint\":\"Rewrite the given line in slope-intercept form, then take the negative reciprocal.\",\"explanation\":\"The first road has equation y = (\\u22124\\\/3)x + 6 and slope \\u22124\\\/3. A perpendicular line has slope 3\\\/4.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"topic\":\"Data Analysis\",\"difficulty\":\"Average\",\"wordProblem\":true,\"prompt\":\"The table shows the number of books read by 10 students during one month. What is the mean number of books read per student?\",\"options\":[\"1.8\",\"2.0\",\"2.4\",\"3.0\"],\"answer\":2,\"drill\":\"Test 2\",\"hint\":\"Multiply each number of books by its frequency, then divide by the number of students.\",\"explanation\":\"The total number of books is 1(2) + 2(4) + 3(2) + 4(2) = 24. The mean is 24\\\/10 = 2.4.\",\"visual\":\"test2-books-table\"},{\"domain\":\"Advanced Math\",\"topic\":\"Nonlinear Equations\",\"difficulty\":\"Average\",\"wordProblem\":true,\"prompt\":\"An engineer uses v = \\u221a(20d) to model speed v, in meters per second, after an object travels d meters. According to the model, what is d when v = 18?\",\"options\":[\"0.9\",\"8.1\",\"16.2\",\"32.4\"],\"answer\":2,\"drill\":\"Test 2\",\"hint\":\"Square both sides before dividing by 20.\",\"explanation\":\"18\\u00b2 = 20d, so d = 324\\\/20 = 16.2 meters.\"},{\"domain\":\"Geometry and Trigonometry\",\"topic\":\"Trigonometry\",\"difficulty\":\"Average\",\"wordProblem\":true,\"prompt\":\"A 10-meter ladder rests against a vertical wall on level ground, forming a right triangle. The ladder makes angle \\u03b8 with the ground, as shown. If cos \\u03b8 = 3\\\/5, how high up the wall does the ladder reach, in meters?\",\"options\":[\"6\",\"8\",\"10\",\"40\\\/3\"],\"answer\":1,\"drill\":\"Test 2\",\"hint\":\"Use cosine to find the distance along the ground, then use the Pythagorean theorem.\",\"explanation\":\"The horizontal distance is 10(3\\\/5) = 6 meters. The vertical height is \\u221a(10\\u00b2 \\u2212 6\\u00b2) = \\u221a64 = 8 meters.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/assessment-print\\\/test2-ladder.svg\"},{\"domain\":\"Algebra\",\"topic\":\"Functions\",\"difficulty\":\"Hard\",\"wordProblem\":true,\"prompt\":\"A delivery company calculates a base charge using f(m) = 4m + 18, where m is the distance in kilometers. It then applies a 10% surcharge to the entire base charge, so the final charge is g(f(m)), where g(x) = 1.1x. If the final charge is 63.80 AED, what is m?\",\"options\":[\"8.0\",\"10.0\",\"11.45\",\"14.5\"],\"answer\":1,\"drill\":\"Test 2\",\"hint\":\"Undo the surcharge before solving for the distance.\",\"explanation\":\"1.1(4m + 18) = 63.80. Dividing by 1.1 gives 4m + 18 = 58, so m = 10 kilometers.\"},{\"domain\":\"Algebra\",\"topic\":\"Linear Equations\",\"difficulty\":\"Hard\",\"wordProblem\":true,\"prompt\":\"A printing company offers Plan A with a 180 AED setup fee and 0.45 AED per page. Plan B has a 60 AED setup fee and costs 0.75 AED per page. A club has a budget of 450 AED. What is the greatest whole number of pages it can print using Plan A while paying strictly less than it would under Plan B?\",\"options\":[\"399\",\"400\",\"520\",\"600\"],\"answer\":3,\"drill\":\"Test 2\",\"hint\":\"Use both the budget limit and the condition that Plan A costs less than Plan B.\",\"explanation\":\"The comparison 180 + 0.45n \\u003C 60 + 0.75n gives n \\u003E 400. The budget 180 + 0.45n \\u2264 450 gives n \\u2264 600. Thus 600 is the greatest permitted number; Plan A costs 450 AED and Plan B costs 510 AED.\"},{\"domain\":\"Advanced Math\",\"topic\":\"Nonlinear Equations\",\"difficulty\":\"Hard\",\"wordProblem\":true,\"prompt\":\"A theater sells 210 tickets when the ticket price is 30 AED. For each 2 AED increase in price, it expects to sell 10 fewer tickets. If x is the number of 2 AED increases, revenue is R(x) = (30 + 2x)(210 \\u2212 10x). What ticket price, in AED, maximizes the predicted revenue?\",\"options\":[\"30\",\"34\",\"36\",\"42\"],\"answer\":2,\"drill\":\"Test 2\",\"hint\":\"Expand the revenue function and find the vertex.\",\"explanation\":\"R(x) = \\u221220x\\u00b2 + 120x + 6,300 has its maximum at x = \\u2212120\\\/(2(\\u221220)) = 3. The corresponding price is 30 + 2(3) = 36 AED.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"topic\":\"Percentages \\u0026 Ratios\",\"difficulty\":\"Hard\",\"wordProblem\":true,\"prompt\":\"A laptop is discounted by 20%. A 5% sales tax is then applied to the discounted price. The final payment is 1,764 AED. What was the original price, in AED?\",\"options\":[\"1,680\",\"2,100\",\"2,205\",\"2,352\"],\"answer\":1,\"drill\":\"Test 2\",\"hint\":\"The two successive multipliers are 0.80 and 1.05.\",\"explanation\":\"If p is the original price, then 0.80(1.05)p = 1,764. Thus 0.84p = 1,764 and p = 2,100 AED.\"},{\"domain\":\"Geometry and Trigonometry\",\"topic\":\"Coordinate Geometry\",\"difficulty\":\"Hard\",\"wordProblem\":true,\"prompt\":\"On a coordinate map, a circular park has endpoints of a diameter at (\\u22122, 3) and (10, 19). Each coordinate unit represents 1 meter. Which equation represents the boundary of the park?\",\"options\":[\"(x \\u2212 4)\\u00b2 + (y \\u2212 11)\\u00b2 = 100\",\"(x \\u2212 6)\\u00b2 + (y \\u2212 8)\\u00b2 = 100\",\"(x \\u2212 4)\\u00b2 + (y \\u2212 11)\\u00b2 = 400\",\"(x + 4)\\u00b2 + (y + 11)\\u00b2 = 100\"],\"answer\":0,\"drill\":\"Test 2\",\"hint\":\"Find the midpoint of the diameter and half its length.\",\"explanation\":\"The center is ((\\u22122 + 10)\\\/2, (3 + 19)\\\/2) = (4, 11). The diameter is \\u221a(12\\u00b2 + 16\\u00b2) = 20, so the radius is 10. Hence (x \\u2212 4)\\u00b2 + (y \\u2212 11)\\u00b2 = 100.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"topic\":\"Data Analysis\",\"difficulty\":\"Hard\",\"wordProblem\":true,\"prompt\":\"A least-squares regression model for a delivery company is C = 7.5d + 42, where d is distance in kilometers and C is predicted cost in AED. One delivery has an actual cost of 192 AED and a residual of \\u221215 AED. Using residual = actual cost \\u2212 predicted cost, what is the distance of this delivery, in kilometers?\",\"options\":[\"18\",\"20\",\"22\",\"26\"],\"answer\":2,\"drill\":\"Test 2\",\"hint\":\"First determine the predicted cost, then substitute it into the regression equation.\",\"explanation\":\"\\u221215 = 192 \\u2212 C, so the predicted cost is 207 AED. Then 207 = 7.5d + 42, giving d = 165\\\/7.5 = 22 kilometers.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A gallery tracked the number of tickets sold at several prices. Sales followed the pattern in the table. About how much more revenue comes from 300 tickets priced at $42 than from 300 tickets priced at $18?\",\"options\":[\"$4,800\",\"$7,200\",\"$9,000\",\"$12,600\"],\"answer\":1,\"drill\":\"Test 3\",\"hint\":\"Multiply each price by the same number of tickets, then subtract.\",\"explanation\":\"The difference is 300($42 \\u2212 $18) = $7,200.\",\"visual\":\"test3-revenue-table\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A workshop packages orders in two daily shifts. According to the table, which pair of days has the largest combined output?\",\"options\":[\"Monday and Thursday\",\"Tuesday and Thursday\",\"Wednesday and Friday\",\"Tuesday and Friday\"],\"answer\":2,\"drill\":\"Test 3\",\"hint\":\"Add the two shift counts for each day and compare the totals.\",\"explanation\":\"The daily totals are 540, 575, 605, 560, and 600. Wednesday and Friday total 1,205, the largest of the listed pairs.\",\"visual\":\"test3-shift-table\"},{\"domain\":\"Advanced Math\",\"prompt\":\"Positive values a, b, and c satisfy a = \\u221a(b\\\/c). Which expression equals c?\",\"options\":[\"b\\\/a\\u00b2\",\"a\\u00b2\\\/b\",\"\\u221a(b\\\/a)\",\"b\\\/a\"],\"answer\":0,\"drill\":\"Test 3\",\"hint\":\"Square both sides before isolating c.\",\"explanation\":\"a\\u00b2 = b\\\/c, so a\\u00b2c = b and c = b\\\/a\\u00b2.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"Solve 9\\u221ax \\u2212 13 = 41 for x.\",\"options\":[\"6\",\"9\",\"36\",\"81\"],\"answer\":2,\"drill\":\"Test 3\",\"hint\":\"Isolate the radical and then square.\",\"explanation\":\"9\\u221ax = 54, so \\u221ax = 6 and x = 36.\"},{\"domain\":\"Algebra\",\"prompt\":\"If 30\\\/(t + 8) = 6\\\/t, what is t\\\/5?\",\"options\":[\"1\\\/4\",\"2\\\/5\",\"1\\\/2\",\"2\"],\"answer\":1,\"drill\":\"Test 3\",\"hint\":\"Cross multiply, and check that t is not zero.\",\"explanation\":\"30t = 6(t + 8), so 24t = 48 and t = 2. Thus t\\\/5 = 2\\\/5.\"},{\"domain\":\"Algebra\",\"prompt\":\"Which system describes the same line twice and therefore has infinitely many solutions?\",\"options\":[\"y = 3x + 4; 2y = 6x + 8\",\"y = 3x + 4; y = 3x \\u2212 4\",\"x = 3; y = 4\",\"y = 2x + 3; y = \\u22122x + 3\"],\"answer\":0,\"drill\":\"Test 3\",\"hint\":\"Equivalent equations have the same slope and intercept.\",\"explanation\":\"Dividing 2y = 6x + 8 by 2 gives y = 3x + 4, exactly the first line.\"},{\"domain\":\"Algebra\",\"prompt\":\"Which inequality is equivalent to \\u22125x + 9 \\u2265 29?\",\"options\":[\"x \\u2265 \\u22124\",\"x \\u2264 \\u22124\",\"x \\u2265 4\",\"x \\u2264 4\"],\"answer\":1,\"drill\":\"Test 3\",\"hint\":\"Dividing by a negative reverses the inequality sign.\",\"explanation\":\"\\u22125x \\u2265 20, so x \\u2264 \\u22124.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"If (x \\u2212 7)\\u00b2 = (x + 1)\\u00b2, what is x?\",\"options\":[\"\\u22124\",\"\\u22123\",\"3\",\"8\"],\"answer\":2,\"drill\":\"Test 3\",\"hint\":\"Subtract the squares and factor.\",\"explanation\":\"(x \\u2212 7)\\u00b2 \\u2212 (x + 1)\\u00b2 = [\\u22128][2x \\u2212 6] = 0, so x = 3.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"For 2x\\u00b2 \\u2212 14x + 20 = 0, what is the product of the two solutions?\",\"options\":[\"5\",\"7\",\"10\",\"20\"],\"answer\":2,\"drill\":\"Test 3\",\"hint\":\"For ax\\u00b2 + bx + c = 0, the product of the roots is c\\\/a.\",\"explanation\":\"The root product is 20\\\/2 = 10; indeed the roots are 2 and 5.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The equation 3x\\u00b2 \\u2212 18x + k = 0 has exactly one real solution. What is k?\",\"options\":[\"9\",\"18\",\"27\",\"36\"],\"answer\":2,\"drill\":\"Test 3\",\"hint\":\"A quadratic has one repeated root when its discriminant is zero.\",\"explanation\":\"(\\u221218)\\u00b2 \\u2212 4(3)(k) = 0 gives 324 \\u2212 12k = 0, so k = 27.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"An account earns 5% interest compounded once per year. What is the least whole number of years needed for its balance to grow by at least 50%?\",\"options\":[\"7\",\"8\",\"9\",\"10\"],\"answer\":2,\"drill\":\"Test 3\",\"hint\":\"Find the smallest n with 1.05\\u207f \\u2265 1.50.\",\"explanation\":\"1.05\\u2078 \\u2248 1.477 \\u003C 1.50 and 1.05\\u2079 \\u2248 1.551 \\u2265 1.50, so 9 years.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"A lab culture has 3,600 cells after 4 equal growth periods. Its size follows N = N\\u2080(3)^(t\\\/2), where t is the number of periods. How many cells were present when t = 0?\",\"options\":[\"300\",\"400\",\"600\",\"1,200\"],\"answer\":1,\"drill\":\"Test 3\",\"hint\":\"At t = 4, the multiplier is 3\\u00b2.\",\"explanation\":\"3,600 = N\\u2080\\u00b73\\u00b2 = 9N\\u2080, so N\\u2080 = 400.\"},{\"domain\":\"Algebra\",\"prompt\":\"If u and v are the solutions of |3x \\u2212 5| = 11, what is |u + v|?\",\"options\":[\"10\\\/3\",\"5\\\/3\",\"16\\\/3\",\"22\\\/3\"],\"answer\":0,\"drill\":\"Test 3\",\"hint\":\"Solve both the positive and negative cases.\",\"explanation\":\"3x \\u2212 5 = 11 gives 16\\\/3, and 3x \\u2212 5 = \\u221211 gives \\u22122. Their sum is 10\\\/3, whose absolute value is 10\\\/3.\"},{\"domain\":\"Algebra\",\"prompt\":\"Which ordered pair satisfies both 3x + y = 11 and 2x \\u2212 y = 4?\",\"options\":[\"(2, 5)\",\"(3, 2)\",\"(4, \\u22121)\",\"(5, \\u22124)\"],\"answer\":1,\"drill\":\"Test 3\",\"hint\":\"Add the two equations to eliminate y.\",\"explanation\":\"5x = 15, so x = 3. Then y = 11 \\u2212 9 = 2.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A farm stand sold 80% of the fruit cups it prepared and sold 144 cups. How many cups did it prepare?\",\"options\":[\"160\",\"172\",\"180\",\"224\"],\"answer\":2,\"drill\":\"Test 3\",\"hint\":\"Divide the amount sold by 0.80.\",\"explanation\":\"144\\\/0.80 = 180 fruit cups.\"},{\"domain\":\"Algebra\",\"prompt\":\"Simplify 4\\\/7 + z\\\/(2 \\u2212 z).\",\"options\":[\"(8 + 3z)\\\/(7(2 \\u2212 z))\",\"(8 + 7z)\\\/(7(2 \\u2212 z))\",\"(4 + z)\\\/(9 \\u2212 z)\",\"(8 \\u2212 3z)\\\/(7(2 \\u2212 z))\"],\"answer\":0,\"drill\":\"Test 3\",\"hint\":\"Use the common denominator 7(2 \\u2212 z).\",\"explanation\":\"[4(2 \\u2212 z) + 7z]\\\/[7(2 \\u2212 z)] = (8 + 3z)\\\/[7(2 \\u2212 z)], for z \\u2260 2.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"During a sale, a desk is marked down 20%. A 5% tax is added to the sale price. If the final charge is T dollars, which expression gives the desk\\u2019s original price?\",\"options\":[\"T\\\/(0.8 \\u00d7 1.05)\",\"T(0.8)(1.05)\",\"T\\\/0.8 + 1.05\",\"T(0.8)\\\/1.05\"],\"answer\":0,\"drill\":\"Test 3\",\"hint\":\"Undo both percentage multipliers.\",\"explanation\":\"T = original price \\u00d7 0.8 \\u00d7 1.05, so original price = T\\\/(0.8 \\u00d7 1.05).\"},{\"domain\":\"Algebra\",\"prompt\":\"A service charges the same amount for each audiobook. If 48 audiobooks cost d dollars, how much do 10 audiobooks cost?\",\"options\":[\"5d\\\/24\",\"24d\\\/5\",\"d\\\/10\",\"10d\"],\"answer\":0,\"drill\":\"Test 3\",\"hint\":\"Find the cost of one audiobook first.\",\"explanation\":\"Each audiobook costs d\\\/48; ten cost 10d\\\/48 = 5d\\\/24.\"},{\"domain\":\"Algebra\",\"prompt\":\"A supply room has r cartons of pencils and s cartons of markers. The equation 15r + 9s = 231 describes the total items. What does 15 represent?\",\"options\":[\"The total pencils\",\"The total markers\",\"Pencils in each pencil carton\",\"Markers in each marker carton\"],\"answer\":2,\"drill\":\"Test 3\",\"hint\":\"The coefficient of r gives items per pencil carton.\",\"explanation\":\"Since r counts pencil cartons, 15 is the number of pencils in each carton.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"A circle in the coordinate plane is described by x\\u00b2 + 8x + y\\u00b2 \\u2212 6y = 24. What is its radius?\",\"options\":[\"5\",\"6\",\"7\",\"\\u221a24\"],\"answer\":2,\"drill\":\"Test 3\",\"hint\":\"Complete the square in x and y.\",\"explanation\":\"(x + 4)\\u00b2 + (y \\u2212 3)\\u00b2 = 24 + 16 + 9 = 49, so the radius is 7.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The table lists 15 travel times. If the smallest time and the largest time are removed, which measure changes the most?\",\"options\":[\"Mode\",\"Median\",\"Range\",\"Mean\"],\"answer\":2,\"drill\":\"Test 3\",\"hint\":\"Compare the old and new minimum and maximum; the middle values stay in place.\",\"explanation\":\"The range falls from 30 \\u2212 4 = 26 to 21 \\u2212 8 = 13, a change of 13. The median remains 13, the mode remains 13, and the mean changes by less than 1.\",\"visual\":\"test3-travel-table\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"In right triangle PQR, angle Q is 45\\u00b0 and PR is perpendicular to QR. If the hypotenuse PQ has length h, what is the area of the triangle?\",\"options\":[\"h\\u00b2\\\/2\",\"h\\u00b2\\\/4\",\"h\\u00b2\\u221a2\\\/4\",\"h\\u00b2\\u221a2\\\/2\"],\"answer\":1,\"drill\":\"Test 3\",\"hint\":\"The legs of a 45\\u00b0\\u201345\\u00b0\\u201390\\u00b0 triangle both have length h\\\/\\u221a2.\",\"explanation\":\"Area = (1\\\/2)(h\\\/\\u221a2)(h\\\/\\u221a2) = h\\u00b2\\\/4.\"},{\"domain\":\"Algebra\",\"prompt\":\"Which number satisfies 5 \\u003E 4x \\u2212 7?\",\"options\":[\"2\",\"3\",\"4\",\"5\"],\"answer\":0,\"drill\":\"Test 4\",\"hint\":\"Solve the strict inequality before checking the choices.\",\"explanation\":\"4x \\u003C 12, so x \\u003C 3. Only 2 is listed below 3.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"Solve \\u221a(16 \\u2212 3x) = x \\u2212 2.\",\"options\":[\"\\u22124\",\"2\",\"4\",\"7\"],\"answer\":2,\"drill\":\"Test 4\",\"hint\":\"Square both sides, then check the original equation.\",\"explanation\":\"Squaring gives 16 \\u2212 3x = x\\u00b2 \\u2212 4x + 4, or x\\u00b2 \\u2212 x \\u2212 12 = 0. The candidates are 4 and \\u22123. Only x = 4 satisfies the original equation: \\u221a4 = 2.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The scatterplot shows nine observations and a fitted line. For how many observations is the measured y-value greater than the value on the line at the same x?\",\"options\":[\"3\",\"4\",\"5\",\"6\"],\"answer\":1,\"drill\":\"Test 4\",\"hint\":\"Count dots strictly above the line.\",\"explanation\":\"Four dots are above the line, at x = 1, 3, 5, and 7.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test4-original\\\/scatter.svg\"},{\"domain\":\"Advanced Math\",\"prompt\":\"When a = \\u22123, what is a + a\\u00b2 \\u2212 a\\u00b3 + a\\u2074?\",\"options\":[\"60\",\"114\",\"120\",\"\\u2212120\"],\"answer\":1,\"drill\":\"Test 4\",\"hint\":\"Evaluate each power before combining the terms.\",\"explanation\":\"\\u22123 + 9 \\u2212 (\\u221227) + 81 = 114.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"If 16\\u207b\\u00b2 = (1\\\/4)^n, what is n?\",\"options\":[\"2\",\"3\",\"4\",\"8\"],\"answer\":2,\"drill\":\"Test 4\",\"hint\":\"Express 16 as a power of 4.\",\"explanation\":\"16\\u207b\\u00b2 = (4\\u00b2)\\u207b\\u00b2 = 4\\u207b\\u2074 = (1\\\/4)\\u2074, so n = 4.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"If \\u221ax + 15 = 26, find x.\",\"options\":[\"11\",\"22\",\"121\",\"169\"],\"answer\":2,\"drill\":\"Test 4\",\"hint\":\"Isolate \\u221ax, then square.\",\"explanation\":\"\\u221ax = 11, so x = 121.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"For what constant c does cx\\u00b2(5x + 3) equal 70x\\u00b3 + 42x\\u00b2?\",\"options\":[\"7\",\"14\",\"28\",\"35\"],\"answer\":1,\"drill\":\"Test 4\",\"hint\":\"Match the coefficient of x\\u00b3.\",\"explanation\":\"5c = 70, so c = 14. The x\\u00b2 term is then 3(14) = 42.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"A rectangular poster is three times as long as it is wide. Its area is 300 square centimeters. How long is it?\",\"options\":[\"10 cm\",\"20 cm\",\"30 cm\",\"100 cm\"],\"answer\":2,\"drill\":\"Test 4\",\"hint\":\"Let the width be w and use 3w\\u00b2 = 300.\",\"explanation\":\"w\\u00b2 = 100, so w = 10 and the length is 3w = 30 cm.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"If \\u221ax\\\/3 = 2\\u221a3, what is x?\",\"options\":[\"12\",\"27\",\"54\",\"108\"],\"answer\":3,\"drill\":\"Test 4\",\"hint\":\"Multiply by 3 and square the result.\",\"explanation\":\"\\u221ax = 6\\u221a3, so x = (6\\u221a3)\\u00b2 = 108.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"Three donations total $540. The first donation is 50% of the other two donations combined. What is the combined value of the other two?\",\"options\":[\"$180\",\"$270\",\"$360\",\"$405\"],\"answer\":2,\"drill\":\"Test 4\",\"hint\":\"Let S represent the sum of the other two donations.\",\"explanation\":\"S + 0.5S = 540, so 1.5S = 540 and S = $360.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"At a school fair, 60% of visitors choose the art exhibit. Of the remaining visitors, 75% choose the robotics exhibit. What percentage choose neither exhibit?\",\"options\":[\"10%\",\"15%\",\"25%\",\"30%\"],\"answer\":0,\"drill\":\"Test 4\",\"hint\":\"Find the fraction outside art, then the fraction of that group outside robotics.\",\"explanation\":\"40% do not choose art, and 25% of that group do not choose robotics. Their share is 0.40 \\u00d7 0.25 = 10%.\"},{\"domain\":\"Algebra\",\"prompt\":\"A linear function f rises by 15 whenever x increases by 3. If f(\\u22122) = \\u221211, which equation defines f?\",\"options\":[\"f(x) = 5x \\u2212 1\",\"f(x) = 5x + 1\",\"f(x) = 3x \\u2212 5\",\"f(x) = \\u22125x \\u2212 21\"],\"answer\":0,\"drill\":\"Test 4\",\"hint\":\"The slope is the change in f divided by the change in x.\",\"explanation\":\"The slope is 15\\\/3 = 5. Using \\u221211 = 5(\\u22122) + b gives b = \\u22121, so f(x) = 5x \\u2212 1.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"A quantity grows by 20% each week. If its value after one week is 36, which function gives its value after t weeks?\",\"options\":[\"q(t) = 30(1.20)^t\",\"q(t) = 36(1.20)^t\",\"q(t) = 30(0.80)^t\",\"q(t) = 36 + 20t\"],\"answer\":0,\"drill\":\"Test 4\",\"hint\":\"Divide the week-one value by 1.20 to find the initial value.\",\"explanation\":\"The starting value is 36\\\/1.20 = 30; hence q(t) = 30(1.20)^t.\"},{\"domain\":\"Algebra\",\"prompt\":\"Which rule matches the linear values in the table?\",\"options\":[\"f(x) = 7x + 3\",\"f(x) = 7x \\u2212 2\",\"f(x) = 5x + 3\",\"f(x) = 3x + 7\"],\"answer\":0,\"drill\":\"Test 4\",\"hint\":\"The output increases by 7 as x increases by 1.\",\"explanation\":\"The table includes f(0) = 3, and every one-unit increase in x adds 7, so f(x) = 7x + 3.\",\"visual\":\"test4-original-linear-table\"},{\"domain\":\"Algebra\",\"prompt\":\"The line shown is 3x + 4y = 12. Which slope could belong to a line perpendicular to it?\",\"options\":[\"\\u22123\\\/4\",\"3\\\/4\",\"\\u22124\\\/3\",\"4\\\/3\"],\"answer\":3,\"drill\":\"Test 4\",\"hint\":\"First find the displayed line\\u2019s slope, then take the negative reciprocal.\",\"explanation\":\"3x + 4y = 12 becomes y = \\u2212(3\\\/4)x + 3. A perpendicular line has slope 4\\\/3.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test4-original\\\/line.svg\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"Circle A has equation (x + 3)\\u00b2 + (y \\u2212 5)\\u00b2 = 64. Circle B has the same radius, and its center is shifted 4 units left from A. Which equation defines B?\",\"options\":[\"(x \\u2212 1)\\u00b2 + (y \\u2212 5)\\u00b2 = 64\",\"(x + 7)\\u00b2 + (y \\u2212 5)\\u00b2 = 64\",\"(x + 3)\\u00b2 + (y \\u2212 1)\\u00b2 = 64\",\"(x + 7)\\u00b2 + (y \\u2212 5)\\u00b2 = 8\"],\"answer\":1,\"drill\":\"Test 4\",\"hint\":\"Read the original center and subtract 4 from its x-coordinate.\",\"explanation\":\"A is centered at (\\u22123, 5). Moving left gives (\\u22127, 5), with radius 8, so (x + 7)\\u00b2 + (y \\u2212 5)\\u00b2 = 64.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The bar chart summarizes 200 parcels by mass. What is the ratio of parcels in the 6\\u201310 kg group to parcels in the 16\\u201320 kg group?\",\"options\":[\"1:3\",\"2:1\",\"3:1\",\"4:1\"],\"answer\":2,\"drill\":\"Test 4\",\"hint\":\"The ratio of counts is the same as the ratio of the displayed percentages.\",\"explanation\":\"The two percentages are 48% and 16%, so 48:16 simplifies to 3:1.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test4-original\\\/bars.svg\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A club increases its membership by 40% and later loses 25% of the new total. The final membership is what percent of the starting membership?\",\"options\":[\"95%\",\"100%\",\"105%\",\"115%\"],\"answer\":2,\"drill\":\"Test 4\",\"hint\":\"Multiply the growth and decrease factors.\",\"explanation\":\"1.40 \\u00d7 0.75 = 1.05, so the final total is 105% of the original.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"The two right triangles shown are similar, with A corresponding to X. What is cos A?\",\"options\":[\"3\\\/5\",\"4\\\/5\",\"5\\\/4\",\"\\u221a3\\\/2\"],\"answer\":1,\"drill\":\"Test 4\",\"hint\":\"Use the corresponding angle X in the small triangle.\",\"explanation\":\"For angle X, the adjacent leg XY is 8 and the hypotenuse XZ is 10. Thus cos A = cos X = 8\\\/10 = 4\\\/5.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test4-original\\\/triangles.svg\"},{\"domain\":\"Algebra\",\"prompt\":\"A rental shop models daily rentals by r = 820 \\u2212 6p, where p is the daily price in dollars. What does the coefficient \\u22126 mean?\",\"options\":[\"A $6 price rise adds one rental.\",\"A $1 price rise reduces rentals by 6.\",\"A $6 price drop reduces rentals by one.\",\"The shop rents 6 items at a price of $820.\"],\"answer\":1,\"drill\":\"Test 4\",\"hint\":\"The slope is the change in rentals when price increases by one dollar.\",\"explanation\":\"Increasing p by 1 decreases r by 6, so the model predicts 6 fewer rentals for each $1 increase.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"In the same scatterplot, which observation has the greatest positive vertical difference from the fitted line?\",\"options\":[\"(1, 27)\",\"(3, 26)\",\"(5, 20)\",\"(7, 16)\"],\"answer\":1,\"drill\":\"Test 4\",\"hint\":\"Compare each dot\\u2019s height with the line directly beneath it.\",\"explanation\":\"At x = 3, the line predicts 22 while the observed value is 26, a positive difference of 4. The other listed positive differences are 1 or 2.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test4-original\\\/scatter.svg\"},{\"domain\":\"Algebra\",\"prompt\":\"A line perpendicular to 3x + 4y = 12 passes through (0, \\u22122). Which equation describes it?\",\"options\":[\"y = (4\\\/3)x \\u2212 2\",\"y = \\u2212(3\\\/4)x \\u2212 2\",\"y = (3\\\/4)x \\u2212 2\",\"y = \\u2212(4\\\/3)x \\u2212 2\"],\"answer\":0,\"drill\":\"Test 4\",\"hint\":\"Use the perpendicular slope and the given y-intercept.\",\"explanation\":\"The original slope is \\u22123\\\/4, so the perpendicular slope is 4\\\/3. Passing through (0, \\u22122) gives y = (4\\\/3)x \\u2212 2.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test4-original\\\/line.svg\"},{\"domain\":\"Algebra\",\"prompt\":\"The product of two positive numbers u and v is 90. Also, u is five times the square of v. Which system represents these conditions?\",\"options\":[\"uv = 90; u = 5v\\u00b2\",\"u + v = 90; u = 5v\\u00b2\",\"uv = 90; v = 5u\\u00b2\",\"uv = 90; u = 5 + v\\u00b2\"],\"answer\":0,\"drill\":\"Test 5\",\"hint\":\"Translate \\u0027product\\u0027 as multiplication and \\u0027five times the square\\u0027 as 5v\\u00b2.\",\"explanation\":\"The product condition is uv = 90. Five times v squared is 5v\\u00b2, so u = 5v\\u00b2.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"An algae sample follows N(w) = c\\u00b72^w, where w is the number of weeks after collection. If N(4) = 48, how many algae cells were present when the sample was collected?\",\"options\":[\"2\",\"3\",\"12\",\"16\"],\"answer\":1,\"drill\":\"Test 5\",\"hint\":\"At collection, w = 0. First solve 48 = c\\u00b72\\u2074.\",\"explanation\":\"48 = 16c gives c = 3. Since 2\\u2070 = 1, N(0) = 3.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"How many real solutions does |x| = |3x \\u2212 4| have?\",\"options\":[\"Zero\",\"One\",\"Two\",\"Infinitely many\"],\"answer\":2,\"drill\":\"Test 5\",\"hint\":\"Set x equal to both 3x \\u2212 4 and its opposite.\",\"explanation\":\"x = 3x \\u2212 4 gives x = 2; x = \\u2212(3x \\u2212 4) gives x = 1. Both work, so there are two solutions.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"A device worth $1,500 loses 18% of its value each year. Which expression models its value, in dollars, after t years?\",\"options\":[\"1,500(0.18)^t\",\"1,500(0.82)^t\",\"1,500(1.18)^t\",\"1,500 \\u2212 18t\"],\"answer\":1,\"drill\":\"Test 5\",\"hint\":\"The device keeps 100% \\u2212 18% of its value each year.\",\"explanation\":\"Each year its value is multiplied by 0.82, giving 1,500(0.82)^t.\"},{\"domain\":\"Algebra\",\"prompt\":\"What ordered pair (x, y) solves the system 3x + y = 11 and 5x + 2y = 19?\",\"options\":[\"(1, 8)\",\"(2, 5)\",\"(3, 2)\",\"(4, \\u22121)\"],\"answer\":2,\"drill\":\"Test 5\",\"hint\":\"Double the first equation and subtract the second.\",\"explanation\":\"6x + 2y = 22. Subtracting 5x + 2y = 19 gives x = 3, and then y = 11 \\u2212 9 = 2.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"At a school fair, 75% of the handmade bookmarks were sold. If 96 bookmarks were sold, how many had been made?\",\"options\":[\"72\",\"120\",\"128\",\"171\"],\"answer\":2,\"drill\":\"Test 5\",\"hint\":\"The 96 sold bookmarks are three-fourths of the starting amount.\",\"explanation\":\"0.75n = 96, so n = 96\\\/0.75 = 128.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"What is the negative solution of |3x \\u2212 2| \\u2212 7 = 0?\",\"options\":[\"\\u22123\",\"\\u22125\\\/3\",\"5\\\/3\",\"3\"],\"answer\":1,\"drill\":\"Test 5\",\"hint\":\"Solve 3x \\u2212 2 = \\u22127 for the negative branch.\",\"explanation\":\"|3x \\u2212 2| = 7, so 3x \\u2212 2 = \\u22127 or 7. The negative solution is x = \\u22125\\\/3.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A coat is marked down 15%, and then 8% sales tax is charged on the discounted price. If the final amount paid is T dollars, which expression gives the original price?\",\"options\":[\"T\\\/(0.85\\u00b71.08)\",\"T\\\/(0.15\\u00b70.08)\",\"0.85\\u00b71.08T\",\"T\\\/(1.15\\u00b71.08)\"],\"answer\":0,\"drill\":\"Test 5\",\"hint\":\"Final price = original price \\u00d7 0.85 \\u00d7 1.08.\",\"explanation\":\"If P is the original price, T = P(0.85)(1.08). Divide by both factors to obtain P = T\\\/(0.85\\u00b71.08).\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A wall clock loses p minutes every q hours, where p and q are positive. After how many hours will it have lost exactly two hours?\",\"options\":[\"2q\\\/p\",\"60q\\\/p\",\"120q\\\/p\",\"120p\\\/q\"],\"answer\":2,\"drill\":\"Test 5\",\"hint\":\"Two hours is 120 minutes. The clock loses p\\\/q minutes per hour.\",\"explanation\":\"At a rate of p\\\/q minutes lost per hour, losing 120 minutes takes 120 \\u00f7 (p\\\/q) = 120q\\\/p hours.\"},{\"domain\":\"Algebra\",\"prompt\":\"A craft kit contains r small packets and s large packets. The equation 9r + 14s = 230 gives the total number of beads. What does 14 represent?\",\"options\":[\"The number of large packets\",\"The number of beads in each large packet\",\"The number of beads in each small packet\",\"The total number of beads\"],\"answer\":1,\"drill\":\"Test 5\",\"hint\":\"In 14s, s counts packets, so 14 is the amount in each one.\",\"explanation\":\"The term 14s counts beads from s large packets; each large packet contains 14 beads.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A machine cuts 18 labels in x minutes at a constant rate. How many labels does it cut in y hours?\",\"options\":[\"18y\\\/x\",\"18xy\\\/60\",\"60y\\\/(18x)\",\"1,080y\\\/x\"],\"answer\":3,\"drill\":\"Test 5\",\"hint\":\"Convert y hours to 60y minutes before multiplying by the rate.\",\"explanation\":\"The rate is 18\\\/x labels per minute. In 60y minutes the machine cuts (18\\\/x)(60y) = 1,080y\\\/x labels.\"},{\"domain\":\"Algebra\",\"prompt\":\"A shipment contains 40 packages, each weighing either 6 ounces or 18 ounces. The combined weight is less than 480 ounces. What is the greatest possible number of 18-ounce packages?\",\"options\":[\"18\",\"19\",\"20\",\"21\"],\"answer\":1,\"drill\":\"Test 5\",\"hint\":\"Let b be the number of 18-ounce packages; the other 40 \\u2212 b weigh 6 ounces each.\",\"explanation\":\"18b + 6(40 \\u2212 b) \\u003C 480 gives 12b \\u003C 240, so b \\u003C 20. The greatest whole number is 19.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"For what value of a does ax\\u00b2 \\u2212 10x + 5 = 0 have exactly one distinct real solution?\",\"options\":[\"2\",\"4\",\"5\",\"10\"],\"answer\":2,\"drill\":\"Test 5\",\"hint\":\"Set the discriminant equal to zero.\",\"explanation\":\"The discriminant is (\\u221210)\\u00b2 \\u2212 4(a)(5) = 100 \\u2212 20a. Setting it to zero gives a = 5.\"},{\"domain\":\"Algebra\",\"prompt\":\"A station sold 360 gallons of fuel for $1,196. Regular fuel cost $3.10 per gallon and premium cost $3.50 per gallon. How many gallons of premium fuel were sold?\",\"options\":[\"160\",\"180\",\"200\",\"240\"],\"answer\":2,\"drill\":\"Test 5\",\"hint\":\"Let p be premium gallons and compare the revenue with 360 gallons all priced at $3.10.\",\"explanation\":\"All regular fuel would bring in $1,116. The extra $80 comes from a $0.40 difference per premium gallon, so p = 80\\\/0.40 = 200.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The value of y is divided by 3 each time x increases by 1. If y = \\u221248 when x = 0, which equation gives y in terms of x?\",\"options\":[\"y = \\u221248(3)^x\",\"y = \\u221248(1\\\/3)^x\",\"y = \\u221248 \\u2212 x\\\/3\",\"y = \\u221216x\"],\"answer\":1,\"drill\":\"Test 5\",\"hint\":\"Use the value at x = 0 as the initial value and 1\\\/3 as the multiplier.\",\"explanation\":\"The initial value is \\u221248 and each unit increase in x multiplies y by 1\\\/3, so y = \\u221248(1\\\/3)^x.\"},{\"domain\":\"Algebra\",\"prompt\":\"The system gx \\u2212 hy = 84 and 5x + 4y = 12 has infinitely many solutions, where g and h are constants. What is gh?\",\"options\":[\"\\u2212980\",\"\\u2212140\",\"140\",\"980\"],\"answer\":0,\"drill\":\"Test 5\",\"hint\":\"The first equation must be seven times the second.\",\"explanation\":\"Multiplying 5x + 4y = 12 by 7 gives 35x + 28y = 84. Thus g = 35 and \\u2212h = 28, so h = \\u221228 and gh = \\u2212980.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The function f is defined by f(x) = \\u22122x\\u00b2 + 12x + 5. For what value of x does f reach its maximum?\",\"options\":[\"\\u22123\",\"2\",\"3\",\"5\"],\"answer\":2,\"drill\":\"Test 5\",\"hint\":\"For a quadratic ax\\u00b2 + bx + c, the vertex occurs at x = \\u2212b\\\/(2a).\",\"explanation\":\"Here x = \\u221212\\\/[2(\\u22122)] = 3. Because the leading coefficient is negative, the vertex is a maximum.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"What is the radius of the circle with equation 2x\\u00b2 \\u2212 8x + 2y\\u00b2 + 12y = 6?\",\"options\":[\"2\",\"3\",\"4\",\"16\"],\"answer\":2,\"drill\":\"Test 5\",\"hint\":\"Divide by 2, then complete the square in x and y.\",\"explanation\":\"Dividing gives x\\u00b2 \\u2212 4x + y\\u00b2 + 6y = 3. Completing squares: (x \\u2212 2)\\u00b2 + (y + 3)\\u00b2 = 16, so the radius is 4.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A jar contains 9 blue tiles and 15 tiles of other colors. If one tile is chosen at random, what is the probability it is blue?\",\"options\":[\"3\\\/8\",\"3\\\/5\",\"9\\\/15\",\"5\\\/8\"],\"answer\":0,\"drill\":\"Test 5\",\"hint\":\"Divide the number of blue tiles by the total number of tiles.\",\"explanation\":\"There are 24 tiles, so P(blue) = 9\\\/24 = 3\\\/8.\"},{\"domain\":\"Algebra\",\"prompt\":\"Which equation must be paired with 6x \\u2212 9y = 15 to form a system with infinitely many solutions?\",\"options\":[\"2x \\u2212 3y = 5\",\"2x \\u2212 3y = 15\",\"6x + 9y = 15\",\"3x \\u2212 9y = 15\"],\"answer\":0,\"drill\":\"Test 5\",\"hint\":\"An equation for the same line has all three terms multiplied by the same factor.\",\"explanation\":\"Dividing 6x \\u2212 9y = 15 by 3 gives 2x \\u2212 3y = 5, the same line and infinitely many common points.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"What is the maximum value of h(x) = \\u2212x\\u00b2 + 8x \\u2212 7?\",\"options\":[\"4\",\"7\",\"9\",\"16\"],\"answer\":2,\"drill\":\"Test 5\",\"hint\":\"Find the vertex at x = \\u2212b\\\/(2a), then evaluate h.\",\"explanation\":\"The vertex is at x = \\u22128\\\/[2(\\u22121)] = 4. Evaluating gives h(4) = \\u221216 + 32 \\u2212 7 = 9.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A bag has 11 green counters and 9 counters that are not green. What is the probability of drawing a counter that is not green?\",\"options\":[\"9\\\/20\",\"11\\\/20\",\"9\\\/11\",\"11\\\/9\"],\"answer\":0,\"drill\":\"Test 5\",\"hint\":\"There are 20 counters altogether.\",\"explanation\":\"The favorable count is 9 and the total is 11 + 9 = 20, so the probability is 9\\\/20.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The function f is defined by f(x) = x\\u00b2 \\u2212 k. If f(\\u22123) = 13, what is k?\",\"options\":[\"\\u22124\",\"4\",\"9\",\"22\"],\"answer\":0,\"drill\":\"Test 6\",\"hint\":\"Substitute \\u22123, then solve 9 \\u2212 k = 13.\",\"explanation\":\"f(\\u22123) = 9 \\u2212 k = 13, so k = \\u22124.\"},{\"domain\":\"Algebra\",\"prompt\":\"A landscaping crew plants 72 trees each day. At this constant rate, which function gives the total trees T planted in d days?\",\"options\":[\"T(d) = 72 + d\",\"T(d) = 72d\",\"T(d) = d\\\/72\",\"T(d) = 72 \\u2212 d\"],\"answer\":1,\"drill\":\"Test 6\",\"hint\":\"Multiply the daily rate by the number of days.\",\"explanation\":\"In d days the crew plants 72 \\u00d7 d = 72d trees.\"},{\"domain\":\"Algebra\",\"prompt\":\"The table gives three pairs (x, y). Which equation fits every row?\",\"options\":[\"y = 2x + 5\",\"y = 3x \\u2212 5\",\"y = 3x + 5\",\"y = 5x + 3\"],\"answer\":2,\"drill\":\"Test 6\",\"hint\":\"The row x = 0 gives the intercept. Then find the change in y per unit x.\",\"explanation\":\"At x = 0, y = 5. From x = 0 to x = 3, y rises 9, so the slope is 3 and y = 3x + 5.\",\"visual\":\"test6-linear-table\"},{\"domain\":\"Algebra\",\"prompt\":\"Line \\u2113 passes through (1, 2) and (5, 10). Which equation could represent a line perpendicular to \\u2113?\",\"options\":[\"2x + y = 9\",\"x + 2y = 9\",\"x \\u2212 2y = 9\",\"2x \\u2212 y = 9\"],\"answer\":1,\"drill\":\"Test 6\",\"hint\":\"Find \\u2113\\u0027s slope, then use its negative reciprocal.\",\"explanation\":\"The slope of \\u2113 is (10 \\u2212 2)\\\/(5 \\u2212 1) = 2. A perpendicular line has slope \\u22121\\\/2; x + 2y = 9 has that slope.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"For positive c, f(x) = 2c^x. If f(3) = 54, what is f(2)?\",\"options\":[\"6\",\"12\",\"18\",\"27\"],\"answer\":2,\"drill\":\"Test 6\",\"hint\":\"First solve 2c\\u00b3 = 54.\",\"explanation\":\"c\\u00b3 = 27 and c = 3. Therefore f(2) = 2(3\\u00b2) = 18.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The graph of a cubic function g is shown. For how many distinct x-values is g(x) = 0?\",\"options\":[\"Zero\",\"One\",\"Two\",\"Three\"],\"answer\":3,\"drill\":\"Test 6\",\"hint\":\"Count the places where the curve meets the x-axis.\",\"explanation\":\"The curve crosses the x-axis at x = \\u22122, 0, and 2, so g(x) = 0 has three distinct solutions.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test6-original\\\/cubic-roots.svg\"},{\"domain\":\"Algebra\",\"prompt\":\"Line \\u2113 passes through (1, 4) and (3, 10). Line m has slope \\u22122 and x-intercept 3. What is the y-coordinate of their intersection?\",\"options\":[\"2\",\"4\",\"6\",\"8\"],\"answer\":1,\"drill\":\"Test 6\",\"hint\":\"Write each line in slope-intercept form, then equate them.\",\"explanation\":\"Line \\u2113 is y = 3x + 1. Line m is y = \\u22122x + 6. Setting them equal gives x = 1 and y = 4.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"Let g(x) = 3x\\u00b2 + 5x and h(x) = x\\u00b2 \\u2212 kx. If g(x)h(x) = 3x\\u2074 \\u2212 7x\\u00b3 \\u2212 20x\\u00b2, what is k?\",\"options\":[\"2\",\"3\",\"4\",\"5\"],\"answer\":2,\"drill\":\"Test 6\",\"hint\":\"The x\\u00b3 coefficient of the product is 5 \\u2212 3k.\",\"explanation\":\"Expanding gives 3x\\u2074 + (5 \\u2212 3k)x\\u00b3 \\u2212 5kx\\u00b2. Thus 5 \\u2212 3k = \\u22127 and k = 4.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"An equipment lease costs $240 the first year and rises by 25% at the start of each following year. What is the total cost for the first three years?\",\"options\":[\"$720\",\"$855\",\"$915\",\"$960\"],\"answer\":2,\"drill\":\"Test 6\",\"hint\":\"Calculate each year\\u0027s price before adding.\",\"explanation\":\"The three yearly payments are $240, $300, and $375. Their sum is $915.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The mean of four lab scores is 74. After a fifth score of 89 is added, what is the new mean?\",\"options\":[\"75\",\"77\",\"78\",\"81\"],\"answer\":1,\"drill\":\"Test 6\",\"hint\":\"The first four scores have a total of 4 \\u00d7 74.\",\"explanation\":\"The new total is 296 + 89 = 385, and 385\\\/5 = 77.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The table lists a sensor\\u0027s operating times in hours:minutes. What is the mean time for the first five trials?\",\"options\":[\"56 minutes\",\"58 minutes\",\"60 minutes\",\"1 hour 4 minutes\"],\"answer\":2,\"drill\":\"Test 6\",\"hint\":\"Convert each of the first five entries to minutes.\",\"explanation\":\"The first five times are 68, 63, 58, 56, and 55 minutes. Their total is 300 minutes, so their mean is 60 minutes.\",\"visual\":\"test6-time-table\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The table shows mineral amounts for five samples of each rock type. How much greater is the median for granite than the median for limestone?\",\"options\":[\"0.22 mg\",\"0.30 mg\",\"0.32 mg\",\"0.40 mg\"],\"answer\":2,\"drill\":\"Test 6\",\"hint\":\"Sort each row\\u0027s five measurements and compare their middle values.\",\"explanation\":\"Granite\\u0027s ordered values have median 0.72 mg; limestone\\u0027s have median 0.40 mg. The difference is 0.32 mg.\",\"visual\":\"test6-rock-table\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The table organizes 240 students by year and elective. If one student is chosen at random, what is the probability that the student is a junior in Robotics?\",\"options\":[\"3\\\/40\",\"3\\\/20\",\"9\\\/40\",\"1\\\/4\"],\"answer\":0,\"drill\":\"Test 6\",\"hint\":\"Use the Junior\\u2013Robotics cell over the grand total.\",\"explanation\":\"There are 18 juniors in Robotics out of 240 students, giving 18\\\/240 = 3\\\/40.\",\"visual\":\"test6-elective-table\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"A square has perimeter 36 units. What is the length of its diagonal?\",\"options\":[\"6\\u221a2\",\"9\",\"9\\u221a2\",\"18\"],\"answer\":2,\"drill\":\"Test 6\",\"hint\":\"Divide the perimeter by 4, then use the Pythagorean theorem.\",\"explanation\":\"Each side is 9. The diagonal is \\u221a(9\\u00b2 + 9\\u00b2) = 9\\u221a2.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"A sphere has volume 288\\u03c0 cubic units. What is its surface area? Use V = (4\\\/3)\\u03c0r\\u00b3 and A = 4\\u03c0r\\u00b2.\",\"options\":[\"36\\u03c0\",\"72\\u03c0\",\"108\\u03c0\",\"144\\u03c0\"],\"answer\":3,\"drill\":\"Test 6\",\"hint\":\"Find r\\u00b3 from the volume first.\",\"explanation\":\"(4\\\/3)\\u03c0r\\u00b3 = 288\\u03c0 gives r\\u00b3 = 216 and r = 6. Thus A = 4\\u03c0(6\\u00b2) = 144\\u03c0.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"Triangle T has a base 20% shorter than the length of rectangle R and a height 25% greater than the width of R. The area of T is what percent of the area of R?\",\"options\":[\"40%\",\"50%\",\"80%\",\"100%\"],\"answer\":1,\"drill\":\"Test 6\",\"hint\":\"Use base 0.80L and height 1.25W in the triangle area formula.\",\"explanation\":\"The triangle area is (1\\\/2)(0.80L)(1.25W) = 0.50LW, which is 50% of the rectangle area LW.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"If f(x) = x\\u00b3 \\u2212 5x + 6, what is f(4)?\",\"options\":[\"34\",\"42\",\"50\",\"70\"],\"answer\":2,\"drill\":\"Test 6\",\"hint\":\"Substitute 4 into every term.\",\"explanation\":\"f(4) = 4\\u00b3 \\u2212 5(4) + 6 = 64 \\u2212 20 + 6 = 50.\"},{\"domain\":\"Algebra\",\"prompt\":\"For what x does f(x) = 19 if f(x) = (3\\\/4)x + 4?\",\"options\":[\"12\",\"16\",\"20\",\"28\"],\"answer\":2,\"drill\":\"Test 6\",\"hint\":\"Subtract 4, then multiply by 4\\\/3.\",\"explanation\":\"(3\\\/4)x = 15, so x = 20.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"Let a and b be the two solutions of |\\u22123x + 1| = 7. What is a + b?\",\"options\":[\"\\u22122\\\/3\",\"0\",\"2\\\/3\",\"7\\\/3\"],\"answer\":2,\"drill\":\"Test 6\",\"hint\":\"Solve the positive and negative cases separately.\",\"explanation\":\"\\u22123x + 1 = 7 yields x = \\u22122; \\u22123x + 1 = \\u22127 yields x = 8\\\/3. Their sum is 2\\\/3.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"An account starts with $200 and grows by 5% each year, with no further transactions. What is the least whole number of years needed for the balance to increase by at least 50%?\",\"options\":[\"8\",\"9\",\"10\",\"11\"],\"answer\":1,\"drill\":\"Test 6\",\"hint\":\"Compare 1.05^8 and 1.05^9 with 1.50.\",\"explanation\":\"1.05^8 \\u2248 1.477 \\u003C 1.50, while 1.05^9 \\u2248 1.551 \\u2265 1.50. The least whole number is 9 years.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"What is the product of the two roots of 2x\\u00b2 \\u2212 14x + 10 = 0?\",\"options\":[\"\\u22127\",\"2\",\"5\",\"10\"],\"answer\":2,\"drill\":\"Test 6\",\"hint\":\"For ax\\u00b2 + bx + c = 0, the product of the roots is c\\\/a.\",\"explanation\":\"The root product is 10\\\/2 = 5.\"},{\"domain\":\"Algebra\",\"prompt\":\"A water filter needs 0.6 liter to prime it and uses another 0.08 liter each week. Which equation gives the total water g, in liters, for w weeks?\",\"options\":[\"g = 0.08 + 0.6w\",\"g = 0.6 + 0.08w\",\"g = 0.68w\",\"g = 0.6(0.08 + w)\"],\"answer\":1,\"drill\":\"Test 6\",\"hint\":\"Add the one-time amount to the weekly amount times w.\",\"explanation\":\"The priming requires 0.6 liter once, and w weeks use 0.08w more, so g = 0.6 + 0.08w.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A data set contains three 3s, two 7s, four 12s, and three 20s. Which frequency list represents the data correctly?\",\"options\":[\"3: 3, 7: 2, 12: 4, 20: 3\",\"3: 2, 7: 3, 12: 4, 20: 3\",\"3: 3, 7: 2, 12: 3, 20: 4\",\"3: 3, 7: 7, 12: 12, 20: 20\"],\"answer\":0,\"drill\":\"Test 7\",\"hint\":\"Pair each distinct value with its count.\",\"explanation\":\"The four distinct values have frequencies 3, 2, 4, and 3, respectively.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"Which expression is equivalent to x\\u00b2 + 2x \\u2212 63?\",\"options\":[\"(x \\u2212 9)(x + 7)\",\"(x + 9)(x \\u2212 7)\",\"(x \\u2212 3)(x + 21)\",\"(x + 3)(x \\u2212 21)\"],\"answer\":1,\"drill\":\"Test 7\",\"hint\":\"Find factors of \\u221263 that sum to 2.\",\"explanation\":\"9 and \\u22127 multiply to \\u221263 and sum to 2, giving (x + 9)(x \\u2212 7).\"},{\"domain\":\"Algebra\",\"prompt\":\"A technician assembles 8 bolts per minute for x minutes and 5 washers per minute for y minutes. Which equation states that the technician assembles 245 pieces in all?\",\"options\":[\"8x + 5y = 245\",\"5x + 8y = 245\",\"13xy = 245\",\"8x + 5y = 13\"],\"answer\":0,\"drill\":\"Test 7\",\"hint\":\"Multiply each rate by the corresponding time.\",\"explanation\":\"The technician assembles 8x bolts and 5y washers, for a total of 8x + 5y = 245.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"In triangle PQR, angle P measures 62\\u00b0 and angle Q measures 47\\u00b0. What is the measure of angle R?\",\"options\":[\"61\\u00b0\",\"69\\u00b0\",\"71\\u00b0\",\"109\\u00b0\"],\"answer\":2,\"drill\":\"Test 7\",\"hint\":\"The interior angles total 180\\u00b0.\",\"explanation\":\"R = 180\\u00b0 \\u2212 62\\u00b0 \\u2212 47\\u00b0 = 71\\u00b0.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The graph shows the amount of material awaiting processing after x work shifts. What does the y-intercept represent?\",\"options\":[\"Material processed per shift\",\"The number of shifts needed\",\"Material remaining after all shifts\",\"The initial amount awaiting processing\"],\"answer\":3,\"drill\":\"Test 7\",\"hint\":\"The y-intercept occurs at x = 0 shifts.\",\"explanation\":\"At zero completed shifts, the graph shows 900 kg awaiting processing: the initial amount.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test7-original\\\/remaining-material.svg\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The table classifies 160 reference books by condition and subject. What is the probability that a randomly chosen book is new?\",\"options\":[\"3\\\/10\",\"2\\\/5\",\"3\\\/5\",\"4\\\/5\"],\"answer\":2,\"drill\":\"Test 7\",\"hint\":\"Use the New row total divided by the grand total.\",\"explanation\":\"There are 96 new books of 160 total, so the probability is 96\\\/160 = 3\\\/5.\",\"visual\":\"test7-books-table\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A college has 48,000 students. In a random sample of 4,000, 68% support a proposal, with a 3-percentage-point margin of error. Which could reasonably be the number of all students who support it?\",\"options\":[\"12,640\",\"29,000\",\"32,640\",\"37,500\"],\"answer\":2,\"drill\":\"Test 7\",\"hint\":\"Apply 65% to 71% to the full population.\",\"explanation\":\"The plausible interval is 0.65(48,000) = 31,200 to 0.71(48,000) = 34,080. The listed value 32,640 is in that interval.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"What is the negative solution of 48\\\/a = a \\u2212 4?\",\"options\":[\"\\u22128\",\"2 \\u2212 2\\u221a13\",\"2 + 2\\u221a13\",\"8\"],\"answer\":1,\"drill\":\"Test 7\",\"hint\":\"Multiply by a (which cannot be zero) and solve the quadratic.\",\"explanation\":\"48 = a\\u00b2 \\u2212 4a, so a\\u00b2 \\u2212 4a \\u2212 48 = 0. The quadratic formula gives a = 2 \\u00b1 2\\u221a13; the negative root is 2 \\u2212 2\\u221a13.\"},{\"domain\":\"Algebra\",\"prompt\":\"A weather balloon starts 850 meters above sea level and rises at a constant 62 meters per minute. Which type of function models its altitude over time?\",\"options\":[\"Increasing linear\",\"Increasing exponential\",\"Decreasing linear\",\"Decreasing exponential\"],\"answer\":0,\"drill\":\"Test 7\",\"hint\":\"A fixed increase for each additional minute gives a constant slope.\",\"explanation\":\"Its altitude is 850 + 62t after t minutes, an increasing linear function.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"What is the x-intercept of y = 7\\u00b73^x \\u2212 7?\",\"options\":[\"(\\u22121, 0)\",\"(0, 0)\",\"(1, 0)\",\"(7, 0)\"],\"answer\":1,\"drill\":\"Test 7\",\"hint\":\"Set y to zero, then solve 3^x = 1.\",\"explanation\":\"0 = 7\\u00b73^x \\u2212 7 means 3^x = 1, which occurs at x = 0.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"In parallelogram ABCD, side AB is one-fourth as long as adjacent side AD. If the perimeter is 100 centimeters, what is AB?\",\"options\":[\"8 cm\",\"10 cm\",\"20 cm\",\"40 cm\"],\"answer\":1,\"drill\":\"Test 7\",\"hint\":\"Let AB = s and AD = 4s; the perimeter is 2(s + 4s).\",\"explanation\":\"10s = 100, so AB = s = 10 centimeters.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test7-original\\\/parallelogram.svg\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"A triangle has area 30 square units, base (m + 7) units, and height m units. What positive value of m satisfies these measurements?\",\"options\":[\"4\",\"5\",\"6\",\"12\"],\"answer\":1,\"drill\":\"Test 7\",\"hint\":\"Set (1\\\/2)m(m + 7) = 30.\",\"explanation\":\"m(m + 7) = 60, so m\\u00b2 + 7m \\u2212 60 = (m + 12)(m \\u2212 5) = 0. The positive value is 5.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The table shows a colony\\u0027s size at successive hours. Which model gives the population C after h hours?\",\"options\":[\"C = 6 + 3h\",\"C = 3\\u00b76^h\",\"C = 6\\u00b73^h\",\"C = 6(1.03)^h\"],\"answer\":2,\"drill\":\"Test 7\",\"hint\":\"Read the initial value and the multiplier between rows.\",\"explanation\":\"At h = 0 the population is 6, and it triples every hour. Thus C = 6\\u00b73^h.\",\"visual\":\"test7-colony-table\"},{\"domain\":\"Algebra\",\"prompt\":\"How many intersection points do the lines 9x \\u2212 y = 0 and 18x + 5 = 2y have?\",\"options\":[\"Zero\",\"One\",\"Two\",\"Infinitely many\"],\"answer\":0,\"drill\":\"Test 7\",\"hint\":\"Write both equations as y = mx + b.\",\"explanation\":\"The lines are y = 9x and y = 9x + 5\\\/2. They are parallel and distinct, so there are zero intersections.\"},{\"domain\":\"Algebra\",\"prompt\":\"A game awards 12 points for each silver token and 7 points for each bronze token. How many points would one token of each type award?\",\"options\":[\"12\",\"19\",\"21\",\"84\"],\"answer\":1,\"drill\":\"Test 7\",\"hint\":\"Add the point values for one of each token.\",\"explanation\":\"One silver and one bronze token award 12 + 7 = 19 points.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"A fund begins at $8,000 and loses 30% of its value each year. Which equation gives its value V after t years?\",\"options\":[\"V = 8,000(0.30)^t\",\"V = 8,000(0.70)^t\",\"V = 8,000 \\u2212 0.30t\",\"V = 0.70(8,000)^t\"],\"answer\":1,\"drill\":\"Test 7\",\"hint\":\"A 30% loss leaves 70% each year.\",\"explanation\":\"The annual multiplier is 0.70, so V = 8,000(0.70)^t.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"A right circular cylinder has volume 200\\u03c0 cubic millimeters and height 8 millimeters. What is its diameter?\",\"options\":[\"5 mm\",\"8 mm\",\"10 mm\",\"25 mm\"],\"answer\":2,\"drill\":\"Test 7\",\"hint\":\"Use V = \\u03c0r\\u00b2h to find r, then double it.\",\"explanation\":\"200\\u03c0 = 8\\u03c0r\\u00b2 gives r\\u00b2 = 25 and r = 5. The diameter is 10 millimeters.\"},{\"domain\":\"Algebra\",\"prompt\":\"The lines 6x + 2y = 8 and 13x + 5y = 12 meet at (a, b). What is a?\",\"options\":[\"\\u22124\",\"\\u22122\",\"2\",\"4\"],\"answer\":3,\"drill\":\"Test 7\",\"hint\":\"From the first equation, y = 4 \\u2212 3x. Substitute into the second.\",\"explanation\":\"13x + 5(4 \\u2212 3x) = 12 gives 20 \\u2212 2x = 12, so x = a = 4.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"In right triangle ABC, angle B is 90\\u00b0 and sin A = 5\\\/13. What is sin C?\",\"options\":[\"5\\\/13\",\"12\\\/13\",\"13\\\/12\",\"1\\\/2\"],\"answer\":1,\"drill\":\"Test 7\",\"hint\":\"A and C are complementary; sin C = cos A.\",\"explanation\":\"The missing leg ratio in a 5\\u201312\\u201313 right triangle is 12\\\/13. Hence sin C = cos A = 12\\\/13.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The quadratic g(x) = \\u22123x\\u00b2 + 18x \\u2212 2 reaches its maximum at x = a. What is a?\",\"options\":[\"\\u22123\",\"2\",\"3\",\"6\"],\"answer\":2,\"drill\":\"Test 7\",\"hint\":\"Find the vertex with x = \\u2212b\\\/(2a) using the quadratic coefficients.\",\"explanation\":\"The vertex x-coordinate is \\u221218\\\/[2(\\u22123)] = 3, and the graph opens downward.\"},{\"domain\":\"Algebra\",\"prompt\":\"The line f(x) = (1\\\/4)x + 2 is perpendicular to line g, and g passes through the origin. What is g(3)?\",\"options\":[\"\\u221212\",\"\\u22123\\\/4\",\"3\\\/4\",\"12\"],\"answer\":0,\"drill\":\"Test 7\",\"hint\":\"The perpendicular slope is the negative reciprocal of 1\\\/4.\",\"explanation\":\"The slope of g is \\u22124 and g(0) = 0, so g(x) = \\u22124x. Thus g(3) = \\u221212.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The system y = 3kx\\u00b2 + 4x + 1 and y\\\/5 = \\u2212x has exactly one real solution, where k is positive. What is k?\",\"options\":[\"9\\\/4\",\"3\",\"27\\\/4\",\"27\\\/2\"],\"answer\":2,\"drill\":\"Test 7\",\"hint\":\"Substitute y = \\u22125x and set the quadratic discriminant to zero.\",\"explanation\":\"3kx\\u00b2 + 9x + 1 = 0 has one real root when 9\\u00b2 \\u2212 4(3k)(1) = 0. Thus 81 \\u2212 12k = 0 and k = 27\\\/4.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"Data set A is 4, 6, 9, 9, 12, 20. Data set B contains the same values except 4. How do their means compare?\",\"options\":[\"A has the smaller mean.\",\"They have equal means.\",\"A has the greater mean.\",\"The means cannot be determined.\"],\"answer\":0,\"drill\":\"Test 8\",\"hint\":\"Compare the totals and numbers of values in each set.\",\"explanation\":\"A has sum 60 over 6 values, so its mean is 10. B has sum 56 over 5 values, so its mean is 11.2. A\\u0027s mean is smaller.\"},{\"domain\":\"Algebra\",\"prompt\":\"If x = 9, what is 42 \\u2212 x?\",\"options\":[\"23\",\"31\",\"33\",\"51\"],\"answer\":2,\"drill\":\"Test 8\",\"hint\":\"Substitute the given value of x.\",\"explanation\":\"42 \\u2212 9 = 33.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A tank held 9,000 liters when its drain opened. After 48 seconds it held 1,800 liters. What was the average drainage rate, in liters per second?\",\"options\":[\"48\",\"150\",\"225\",\"7,200\"],\"answer\":1,\"drill\":\"Test 8\",\"hint\":\"Divide the amount lost by the elapsed time.\",\"explanation\":\"The tank lost 9,000 \\u2212 1,800 = 7,200 liters. Dividing by 48 seconds gives 150 liters per second.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"Similar right triangles ABC and DEF have right angles at A and D, and B corresponds to E. If angle E is 58\\u00b0, what is angle C?\",\"options\":[\"32\\u00b0\",\"58\\u00b0\",\"122\\u00b0\",\"148\\u00b0\"],\"answer\":0,\"drill\":\"Test 8\",\"hint\":\"The two acute angles in a right triangle sum to 90\\u00b0.\",\"explanation\":\"Angle B is 58\\u00b0 because it corresponds to E. Therefore C = 90\\u00b0 \\u2212 58\\u00b0 = 32\\u00b0.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test8-original\\\/similar-triangles.svg\"},{\"domain\":\"Algebra\",\"prompt\":\"A club sold b large posters at $420 each and g small posters at $260 each. The equation 420b + 260g = 18,200 describes the revenue. How much more does one large poster cost than one small poster?\",\"options\":[\"$160\",\"$260\",\"$420\",\"$680\"],\"answer\":0,\"drill\":\"Test 8\",\"hint\":\"The coefficients are the prices per poster.\",\"explanation\":\"A large poster costs $420 and a small poster costs $260, so the difference is $160.\"},{\"domain\":\"Algebra\",\"prompt\":\"The system 7x \\u2212 y = \\u221213 and \\u22125x = 15 has solution (x, y). What is x \\u2212 y?\",\"options\":[\"\\u221211\",\"\\u22125\",\"5\",\"11\"],\"answer\":2,\"drill\":\"Test 8\",\"hint\":\"Find x from the second equation, then substitute into the first.\",\"explanation\":\"x = \\u22123. Then \\u221221 \\u2212 y = \\u221213, so y = \\u22128. Hence x \\u2212 y = \\u22123 \\u2212 (\\u22128) = 5.\"},{\"domain\":\"Algebra\",\"prompt\":\"The balance of a community fund is modeled by B(d) = 450d + 9,800, where d is days since donations began. What balance did the fund have just before donations began?\",\"options\":[\"$450\",\"$9,350\",\"$9,800\",\"$10,250\"],\"answer\":2,\"drill\":\"Test 8\",\"hint\":\"Use d = 0 for the starting balance.\",\"explanation\":\"B(0) = 450(0) + 9,800 = $9,800.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"The perimeter of equilateral triangle P is two-fifths the perimeter of equilateral triangle Q. If each side of Q is 15 centimeters, how long is each side of P?\",\"options\":[\"3 cm\",\"5 cm\",\"6 cm\",\"12 cm\"],\"answer\":2,\"drill\":\"Test 8\",\"hint\":\"Equilateral triangles\\u0027 side lengths have the same ratio as their perimeters.\",\"explanation\":\"Each side of P is (2\\\/5)(15) = 6 centimeters.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"If |5x + 10| = 35, which is one possible value of x + 2?\",\"options\":[\"\\u22125\",\"2\",\"5\",\"7\"],\"answer\":3,\"drill\":\"Test 8\",\"hint\":\"Use 5x + 10 = 35 for one solution and then evaluate x + 2.\",\"explanation\":\"The positive case gives x = 5, so x + 2 = 7. The other case gives \\u22127, which is not listed.\"},{\"domain\":\"Algebra\",\"prompt\":\"The table shows values of a linear function g. If g(x) = kx + 6, what is k?\",\"options\":[\"\\u22124\",\"2\",\"4\",\"6\"],\"answer\":2,\"drill\":\"Test 8\",\"hint\":\"Divide the change in output by the corresponding change in input.\",\"explanation\":\"Between x = \\u221220 and x = \\u221215, g(x) rises from \\u221274 to \\u221254, a change of 20 over an input change of 5. Thus k = 4.\",\"visual\":\"test8-linear-table\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The expression (\\u22124x\\u00b3 + 17) + (\\u22129x\\u00b3 \\u2212 5) equals cx\\u00b3 + 12. What is c?\",\"options\":[\"\\u221213\",\"\\u22125\",\"5\",\"13\"],\"answer\":0,\"drill\":\"Test 8\",\"hint\":\"Combine the like x\\u00b3 terms.\",\"explanation\":\"\\u22124x\\u00b3 \\u2212 9x\\u00b3 = \\u221213x\\u00b3, and 17 \\u2212 5 = 12. Therefore c = \\u221213.\"},{\"domain\":\"Algebra\",\"prompt\":\"If 18m\\\/n = 3\\\/(2s), where n and s are nonzero, which equation expresses m in terms of n and s?\",\"options\":[\"m = 12ns\",\"m = n\\\/(12s)\",\"m = 3n\\\/(2s)\",\"m = 12s\\\/n\"],\"answer\":1,\"drill\":\"Test 8\",\"hint\":\"Multiply both sides by n\\\/18 and simplify.\",\"explanation\":\"m = (n\\\/18)(3\\\/(2s)) = n\\\/(12s).\"},{\"domain\":\"Advanced Math\",\"prompt\":\"A cell culture begins with 64,000 cells and half the cells remain after every 3 days. How many cells remain after 12 days?\",\"options\":[\"2,000\",\"4,000\",\"8,000\",\"16,000\"],\"answer\":1,\"drill\":\"Test 8\",\"hint\":\"There are four three-day periods in 12 days.\",\"explanation\":\"64,000(1\\\/2)\\u2074 = 64,000\\\/16 = 4,000 cells.\"},{\"domain\":\"Algebra\",\"prompt\":\"The inequalities x \\u003C 50 and x \\u2212 5y \\u003C 15 have a solution (35, y). Which listed value could be y?\",\"options\":[\"\\u22125\",\"3\",\"4\",\"5\"],\"answer\":3,\"drill\":\"Test 8\",\"hint\":\"Substitute x = 35 into the second inequality.\",\"explanation\":\"35 \\u2212 5y \\u003C 15 implies \\u22125y \\u003C \\u221220, or y \\u003E 4. Only 5 qualifies.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"For positive s and t, which expression equals the ninth root of s\\u2076t\\u2076?\",\"options\":[\"(st)^(2\\\/3)\",\"(st)^(3\\\/2)\",\"(st)^6\",\"(st)^54\"],\"answer\":0,\"drill\":\"Test 8\",\"hint\":\"A ninth root corresponds to a power of 1\\\/9.\",\"explanation\":\"(s\\u2076t\\u2076)^(1\\\/9) = (st)^(6\\\/9) = (st)^(2\\\/3).\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The graph shows a collectible\\u0027s value y, measured in hundreds of dollars, x months after purchase. What does its y-intercept mean?\",\"options\":[\"The purchase value was $3.\",\"The purchase value was $300.\",\"The value rose by $300 in one month.\",\"The value at month 6 was $3,000.\"],\"answer\":1,\"drill\":\"Test 8\",\"hint\":\"The y-intercept gives the value at x = 0 months; account for hundreds of dollars.\",\"explanation\":\"The graph begins at y = 3 when x = 0. Since y is in hundreds of dollars, the purchase value was $300.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test8-original\\\/collectible-value.svg\"},{\"domain\":\"Algebra\",\"prompt\":\"Which system of lines has exactly one real solution?\",\"options\":[\"y = 3; y = 6\",\"y = 2x + 1; y = 2x \\u2212 4\",\"y = 4x \\u2212 2; y = \\u2212x + 3\",\"y = 5x + 2; 2y = 10x + 4\"],\"answer\":2,\"drill\":\"Test 8\",\"hint\":\"Different slopes produce exactly one intersection.\",\"explanation\":\"The lines in the third system have slopes 4 and \\u22121, so they meet at exactly one point. The other systems are distinct parallel lines or the same line.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"A sequence begins at 81, and each following term is one-third of the previous term. Which formula gives its nth term a\\u2099?\",\"options\":[\"a\\u2099 = 81(1\\\/3)^n\",\"a\\u2099 = 81(1\\\/3)^(n\\u22121)\",\"a\\u2099 = (1\\\/3)(81)^(n\\u22121)\",\"a\\u2099 = 81 \\u2212 n\\\/3\"],\"answer\":1,\"drill\":\"Test 8\",\"hint\":\"For a geometric sequence use first term times ratio^(n\\u22121).\",\"explanation\":\"The first term is 81 and common ratio is 1\\\/3, giving a\\u2099 = 81(1\\\/3)^(n\\u22121).\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A music platform\\u0027s catalog decreased by 12% from one year to the next. The new count is m times the old count. What is m?\",\"options\":[\"0.12\",\"0.88\",\"1.12\",\"1.88\"],\"answer\":1,\"drill\":\"Test 8\",\"hint\":\"Subtract the percentage lost from 100%.\",\"explanation\":\"The new count is 100% \\u2212 12% = 88% of the old count, so m = 0.88.\"},{\"domain\":\"Algebra\",\"prompt\":\"A quantity p is allowed to be at most 6 more than four times r. Which inequality gives this limit?\",\"options\":[\"p \\u003C 4r + 6\",\"p \\u2264 4r + 6\",\"p \\u2265 4r + 6\",\"p \\u2264 6r + 4\"],\"answer\":1,\"drill\":\"Test 8\",\"hint\":\"At most includes the upper limit itself.\",\"explanation\":\"The greatest allowed value is 4r + 6, so p \\u2264 4r + 6.\"},{\"domain\":\"Algebra\",\"prompt\":\"The line 4x + 2y = \\u221212 is moved up 3 units. What is the x-coordinate of the x-intercept of the new line?\",\"options\":[\"\\u22123\",\"\\u22123\\\/2\",\"3\\\/2\",\"3\"],\"answer\":1,\"drill\":\"Test 8\",\"hint\":\"Write the old line as y = mx + b, add 3 to b, and set y = 0.\",\"explanation\":\"The old line is y = \\u22122x \\u2212 6. Moving it up 3 gives y = \\u22122x \\u2212 3. At y = 0, x = \\u22123\\\/2.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"Right triangles ABC and DEF are similar, with A corresponding to D. In ABC, AB = 25, BC = 60, and AC = 65; B is the right angle. What is cos D?\",\"options\":[\"5\\\/13\",\"12\\\/13\",\"13\\\/12\",\"13\\\/5\"],\"answer\":0,\"drill\":\"Test 8\",\"hint\":\"D corresponds to A; use A\\u0027s adjacent leg over the hypotenuse.\",\"explanation\":\"cos D = cos A = AB\\\/AC = 25\\\/65 = 5\\\/13.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The table shows books read by students in a month. What is the mean number of books read per student?\",\"options\":[\"4.5\",\"4.8\",\"5.2\",\"6.0\"],\"answer\":1,\"drill\":\"Test 9\",\"hint\":\"Multiply each number of books by its frequency, add the products, and divide by the total frequency.\",\"explanation\":\"The weighted sum is 2(2)+4(3)+6(4)+8(1)=48 books for 10 students. The mean is 4.8.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test9\\\/question-01.svg\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The function g is defined by g(x) = \\u221a(x + 5)\\\/4. For which x is g(x) = 2?\",\"options\":[\"27\",\"43\",\"59\",\"69\"],\"answer\":2,\"drill\":\"Test 9\",\"hint\":\"Isolate the square root, then square both sides.\",\"explanation\":\"\\u221a(x+5)=8, so x+5=64 and x=59.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"Positive a, b, and c satisfy a = \\u221b(b\\\/c). Which expression equals c?\",\"options\":[\"b\\\/a\\u00b3\",\"a\\u00b3\\\/b\",\"\\u221b(b\\\/a)\",\"b\\\/a\"],\"answer\":0,\"drill\":\"Test 9\",\"hint\":\"Cube both sides before isolating c.\",\"explanation\":\"Cubing gives a\\u00b3=b\\\/c. Therefore c=b\\\/a\\u00b3.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"Solve \\u221a(29 \\u2212 5x) = x \\u2212 3.\",\"options\":[\"\\u22124\",\"3\",\"5\",\"8\"],\"answer\":2,\"drill\":\"Test 9\",\"hint\":\"Square both sides, then check the original equation.\",\"explanation\":\"Squaring gives 29\\u22125x=(x\\u22123)\\u00b2, or x\\u00b2\\u2212x\\u221220=0. The roots are 5 and \\u22124. Only 5 satisfies the original equation.\"},{\"domain\":\"Algebra\",\"prompt\":\"The product of positive u and v is 84, and u is seven times the square of v. Which system represents this?\",\"options\":[\"uv=84; u=7v\\u00b2\",\"u+v=84; u=7v\\u00b2\",\"uv=84; v=7u\\u00b2\",\"uv=84; u=7+v\\u00b2\"],\"answer\":0,\"drill\":\"Test 9\",\"hint\":\"Translate product and seven times the square into equations.\",\"explanation\":\"Product means uv=84, and seven times v squared means u=7v\\u00b2.\"},{\"domain\":\"Algebra\",\"prompt\":\"Line \\u2113 passes through (2, 3) and (6, 15). Which equation could represent a line perpendicular to \\u2113?\",\"options\":[\"3x+y=12\",\"x+3y=12\",\"x\\u22123y=12\",\"3x\\u2212y=12\"],\"answer\":1,\"drill\":\"Test 9\",\"hint\":\"Find \\u2113\\u0027s slope, then use its negative reciprocal.\",\"explanation\":\"The slope of \\u2113 is (15\\u22123)\\\/(6\\u22122)=3. The perpendicular slope is \\u22121\\\/3, as in x+3y=12.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A graph shows material awaiting processing after x work shifts. What does the y-intercept represent?\",\"options\":[\"Material processed per shift\",\"The number of shifts needed\",\"Material remaining after all shifts\",\"The initial amount awaiting processing\"],\"answer\":3,\"drill\":\"Test 9\",\"hint\":\"The y-intercept occurs at x = 0 shifts.\",\"explanation\":\"At x=0, no shifts have occurred, so the y-intercept gives the starting amount: 750 kg.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test9\\\/question-07.svg\"},{\"domain\":\"Advanced Math\",\"prompt\":\"If |4x + 12| = 32, which is one possible value of x + 3?\",\"options\":[\"\\u22125\",\"3\",\"8\",\"11\"],\"answer\":2,\"drill\":\"Test 9\",\"hint\":\"Solve both cases of the absolute value equation.\",\"explanation\":\"4(x+3)=32 or \\u221232, so x+3=8 or \\u22128. Choice C is possible.\"},{\"domain\":\"Algebra\",\"prompt\":\"The graph shows the x-intercept and y-intercept of line \\u2113. Which equation represents line \\u2113?\",\"options\":[\"y=2x+8\",\"y=\\u22122x+8\",\"y=\\u2212x\\\/2+8\",\"y=\\u22122x+4\"],\"answer\":1,\"drill\":\"Test 9\",\"hint\":\"Use the two labeled intercepts to calculate the slope.\",\"explanation\":\"The intercepts are (0,8) and (4,0), so the slope is \\u22122 and the equation is y=\\u22122x+8.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test9\\\/question-09.svg\"},{\"domain\":\"Algebra\",\"prompt\":\"A number z satisfies 3\\\/(z + 2) = 1\\\/(z \\u2212 4). What is 3\\\/z?\",\"options\":[\"7\",\"7\\\/3\",\"3\",\"3\\\/7\"],\"answer\":3,\"drill\":\"Test 9\",\"hint\":\"Cross-multiply to solve for z.\",\"explanation\":\"3(z\\u22124)=z+2 gives 2z=14 and z=7. Thus 3\\\/z=3\\\/7.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The equation 2x\\u00b2 \\u2212 16x + k = 0 has exactly one real solution. What is k?\",\"options\":[\"16\",\"24\",\"32\",\"64\"],\"answer\":2,\"drill\":\"Test 9\",\"hint\":\"A quadratic has one repeated root when its discriminant is zero.\",\"explanation\":\"The discriminant must be zero: (\\u221216)\\u00b2\\u22124(2)k=0, so k=32.\"},{\"domain\":\"Algebra\",\"prompt\":\"Which rule matches the linear values in the table?\",\"options\":[\"f(x)=6x+5\",\"f(x)=6x\\u22125\",\"f(x)=5x+6\",\"f(x)=3x+5\"],\"answer\":0,\"drill\":\"Test 9\",\"hint\":\"Find the change in output for each unit of x.\",\"explanation\":\"Each increase of 1 in x raises f(x) by 6, and f(0)=5, so f(x)=6x+5.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test9\\\/question-12.svg\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A coat is marked down 20%, then 6% sales tax is charged. If the final amount is T, which expression gives the original price?\",\"options\":[\"T\\\/(0.80\\u00b71.06)\",\"T\\\/(0.20\\u00b70.06)\",\"0.80\\u00b71.06T\",\"T\\\/(1.20\\u00b71.06)\"],\"answer\":0,\"drill\":\"Test 9\",\"hint\":\"Divide by the discount and tax multipliers.\",\"explanation\":\"The final charge is original price times 0.80 times 1.06. Divide T by that product.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The table lists operating times in hours:minutes. What is the mean time for the first five trials?\",\"options\":[\"60 minutes\",\"62 minutes\",\"64 minutes\",\"68 minutes\"],\"answer\":2,\"drill\":\"Test 9\",\"hint\":\"Convert each of the first five entries to minutes.\",\"explanation\":\"The first five times are 72, 67, 63, 60, and 58 minutes. Their total is 320 minutes, so the mean is 64 minutes.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test9\\\/question-14.svg\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The table shows a colony\\u0027s size at successive hours. Which model gives the population C after h hours?\",\"options\":[\"C=8+2h\",\"C=2\\u00b78^h\",\"C=8\\u00b72^h\",\"C=8(1.02)^h\"],\"answer\":2,\"drill\":\"Test 9\",\"hint\":\"Read the initial value and the multiplier between rows.\",\"explanation\":\"The population starts at 8 and doubles each hour, so C=8\\u00b72^h.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test9\\\/question-15.svg\"},{\"domain\":\"Advanced Math\",\"prompt\":\"For positive s and t, which expression equals the twelfth root of s\\u2078t\\u2078?\",\"options\":[\"(st)^(2\\\/3)\",\"(st)^(3\\\/2)\",\"(st)^8\",\"(st)^96\"],\"answer\":0,\"drill\":\"Test 9\",\"hint\":\"A twelfth root is a power of 1\\\/12.\",\"explanation\":\"(s\\u2078t\\u2078)^(1\\\/12)=(st)^(8\\\/12)=(st)^(2\\\/3).\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The table gives study time x and test score y. Which equation is the best linear regression model?\",\"options\":[\"y=2.5x+5.3\",\"y=3.5x+2.5\",\"y=\\u22122.5x+20.3\",\"y=5x+1\"],\"answer\":0,\"drill\":\"Test 9\",\"hint\":\"A regression line should follow the positive trend and pass near the center of the data.\",\"explanation\":\"For x=1,2,3,4,5 and y=8,10,13,15,18, the least-squares slope is 2.5 and intercept is 5.3.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test9\\\/question-17.svg\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"The system 2x\\u00b2 + y = 4 and 4x \\u2212 y = 2 has two solutions. Which is one y-coordinate?\",\"options\":[\"\\u221212\",\"\\u22122\",\"2\",\"14\"],\"answer\":2,\"drill\":\"Test 9\",\"hint\":\"Solve the linear equation for y, substitute it into the quadratic equation, then find y from either x-value.\",\"explanation\":\"Substitute y=4x\\u22122: x\\u00b2+2x\\u22123=0, so x=1 or \\u22123. When x=1, y=2.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"During a sale, a desk is marked down 15%. An 8% tax is added to the sale price. If the final charge is T, which expression gives the original price?\",\"options\":[\"T\\\/(0.85\\u00d71.08)\",\"T(0.85)(1.08)\",\"T\\\/0.85+1.08\",\"T(0.85)\\\/1.08\"],\"answer\":0,\"drill\":\"Test 9\",\"hint\":\"Undo both percentage multipliers.\",\"explanation\":\"The final charge is original price multiplied by 0.85 and 1.08, so reverse both factors.\"},{\"domain\":\"Algebra\",\"prompt\":\"Daily rentals are modeled by r = 940 \\u2212 8p, where p is the price in dollars. What does \\u22128 mean?\",\"options\":[\"An $8 increase adds one rental.\",\"A $1 increase reduces rentals by 8.\",\"An $8 drop reduces rentals by one.\",\"The shop rents 8 items at $940.\"],\"answer\":1,\"drill\":\"Test 9\",\"hint\":\"The slope is the change in rentals when price increases by one dollar.\",\"explanation\":\"The slope \\u22128 means each $1 increase in price corresponds to eight fewer rentals.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"What is the radius of the circle x\\u00b2 \\u2212 6x + y\\u00b2 + 8y = 0?\",\"options\":[\"3\",\"4\",\"5\",\"25\"],\"answer\":2,\"drill\":\"Test 9\",\"hint\":\"Complete the square in x and y.\",\"explanation\":\"Complete the squares: (x\\u22123)\\u00b2+(y+4)\\u00b2=25. The radius is \\u221a25=5.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"Triangle T has a base 30% shorter than a rectangle\\u0027s length and a height 20% greater than its width. Its area is what percent of the rectangle\\u0027s?\",\"options\":[\"35%\",\"42%\",\"70%\",\"84%\"],\"answer\":1,\"drill\":\"Test 9\",\"hint\":\"Use base 0.70L and height 1.20W in the triangle formula.\",\"explanation\":\"The ratio of areas is (1\\\/2)(0.70)(1.20)=0.42, or 42%.\"},{\"domain\":\"Algebra\",\"prompt\":\"The equation x\\\/y + 5 = 10 \\u2212 3\\\/y relates x and nonzero y. Which expression gives x in terms of y?\",\"options\":[\"x=5y\\u22123\",\"x=5y+3\",\"x=15y\\u22123\",\"x=5\\u22123y\"],\"answer\":0,\"drill\":\"Test 9\",\"hint\":\"Multiply the entire equation by y, then isolate x.\",\"explanation\":\"Multiply by y: x+5y=10y\\u22123, so x=5y\\u22123.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"Which expression is equivalent to x\\u00b2 \\u2212 11x + 24?\",\"options\":[\"(x\\u22122)(x\\u221212)\",\"(x\\u22123)(x\\u22128)\",\"(x+3)(x+8)\",\"(x\\u22124)(x\\u22126)\"],\"answer\":1,\"drill\":\"Test 9\",\"hint\":\"Find two numbers whose product is 24 and whose sum is \\u221211.\",\"explanation\":\"The numbers 3 and 8 multiply to 24 and add to 11, so the factors are (x\\u22123)(x\\u22128).\"},{\"domain\":\"Algebra\",\"prompt\":\"Which system describes the same line twice and has infinitely many solutions?\",\"options\":[\"y=4x+5; 3y=12x+15\",\"y=4x+5; y=4x\\u22125\",\"x=4; y=5\",\"y=3x+4; y=\\u22123x+4\"],\"answer\":0,\"drill\":\"Test 9\",\"hint\":\"Equivalent equations have the same slope and intercept.\",\"explanation\":\"Multiplying y=4x+5 by 3 gives 3y=12x+15, the same line.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The scatterplot shows observations and a fitted line. How many measured y-values are greater than the line\\u0027s predicted value?\",\"options\":[\"3\",\"4\",\"5\",\"6\"],\"answer\":1,\"drill\":\"Test 9\",\"hint\":\"Count dots strictly above the line.\",\"explanation\":\"Count the plotted points vertically above the line of best fit: there are four.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test9\\\/question-26.svg\"},{\"domain\":\"Algebra\",\"prompt\":\"What ordered pair solves 2x + y = 13 and 4x + 3y = 31?\",\"options\":[\"(2,9)\",\"(3,7)\",\"(4,5)\",\"(5,3)\"],\"answer\":2,\"drill\":\"Test 9\",\"hint\":\"Substitute y=13\\u22122x in the second equation.\",\"explanation\":\"From y=13\\u22122x, substitute into 4x+3y=31 to get x=4 and y=5.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The graph of cubic function g is shown. For how many distinct x-values is g(x)=0?\",\"options\":[\"Zero\",\"One\",\"Two\",\"Three\"],\"answer\":3,\"drill\":\"Test 9\",\"hint\":\"Count the places where the curve meets the x-axis.\",\"explanation\":\"The curve meets the x-axis at three distinct points.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test9\\\/question-28.svg\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"In parallelogram ABCD, AB is one-fourth of adjacent side AD. If the perimeter is 120 cm, what is AB?\",\"options\":[\"10 cm\",\"12 cm\",\"24 cm\",\"48 cm\"],\"answer\":1,\"drill\":\"Test 9\",\"hint\":\"Let AB = s and AD = 4s; the perimeter is 2(s + 4s).\",\"explanation\":\"Let AB=s and AD=4s. Then 2(s+4s)=120, so s=12 cm.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test9\\\/question-29.svg\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"Similar right triangles ABC and DEF have right angles at A and D, with B corresponding to E. If angle E is 54\\u00b0, what is angle C?\",\"options\":[\"36\\u00b0\",\"54\\u00b0\",\"126\\u00b0\",\"144\\u00b0\"],\"answer\":0,\"drill\":\"Test 9\",\"hint\":\"The two acute angles in a right triangle sum to 90\\u00b0.\",\"explanation\":\"Angle B equals E=54\\u00b0. The two acute angles in right triangle ABC sum to 90\\u00b0, so C=36\\u00b0.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test9\\\/question-30.svg\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"Parallel lines p and q are intersected by transversal t. If one same-side interior angle is 124\\u00b0, what is x?\",\"options\":[\"54\",\"56\",\"124\",\"126\"],\"answer\":1,\"drill\":\"Test 9\",\"hint\":\"The two labeled angles are same-side interior angles formed by parallel lines.\",\"explanation\":\"Same-side interior angles are supplementary: x=180\\u2212124=56\\u00b0.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test9\\\/question-31.svg\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"After a 25% discount, a jacket costs $84. What was its original price?\",\"options\":[\"$105\",\"$112\",\"$116\",\"$120\"],\"answer\":1,\"drill\":\"Test 9\",\"hint\":\"The sale price is 75% of the original.\",\"explanation\":\"The sale price is 75% of the original, so the original price is 84\\\/0.75=$112.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"If u and v solve |4x \\u2212 7| = 13, what is |u + v|?\",\"options\":[\"7\\\/2\",\"7\\\/4\",\"13\\\/2\",\"13\\\/4\"],\"answer\":0,\"drill\":\"Test 9\",\"hint\":\"Solve both the positive and negative cases.\",\"explanation\":\"The roots are x=5 and x=\\u22123\\\/2. Their sum is 7\\\/2, whose absolute value is 7\\\/2.\"},{\"domain\":\"Algebra\",\"prompt\":\"A linear function rises by 18 whenever x increases by 3. If f(\\u22121)=\\u22128, which equation defines f?\",\"options\":[\"f(x)=6x\\u22122\",\"f(x)=6x+2\",\"f(x)=3x\\u22125\",\"f(x)=\\u22126x\\u221214\"],\"answer\":0,\"drill\":\"Test 9\",\"hint\":\"The slope is the change in f divided by the change in x.\",\"explanation\":\"The slope is 18\\\/3=6. From \\u22128=6(\\u22121)+b, b=\\u22122.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The value of y is divided by 4 each time x increases by 1. If y=\\u221264 at x=0, which equation gives y?\",\"options\":[\"y=\\u221264(4)^x\",\"y=\\u221264(1\\\/4)^x\",\"y=\\u221264\\u2212x\\\/4\",\"y=\\u221216x\"],\"answer\":1,\"drill\":\"Test 9\",\"hint\":\"Use the initial value and 1\\\/4 as the multiplier.\",\"explanation\":\"The initial value is \\u221264 and each step multiplies by 1\\\/4, so y=\\u221264(1\\\/4)^x.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The table lists mineral amounts for five samples of each rock type. How much greater is granite\\u0027s median than limestone\\u0027s?\",\"options\":[\"0.25 mg\",\"0.30 mg\",\"0.35 mg\",\"0.40 mg\"],\"answer\":2,\"drill\":\"Test 9\",\"hint\":\"Sort each row\\u0027s five measurements and compare their middle values.\",\"explanation\":\"Granite sorted is 0.72,0.76,0.79,0.81,0.85, median 0.79. Limestone sorted is 0.40,0.42,0.44,0.46,0.48, median 0.44. Difference: 0.35 mg.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test9\\\/question-36.svg\"},{\"domain\":\"Advanced Math\",\"prompt\":\"A fund begins at $9,000 and loses 25% of its value each year. Which equation gives V after t years?\",\"options\":[\"V=9,000(0.25)^t\",\"V=9,000(0.75)^t\",\"V=9,000\\u22120.25t\",\"V=0.75(9,000)^t\"],\"answer\":1,\"drill\":\"Test 9\",\"hint\":\"A 25% loss leaves 75%.\",\"explanation\":\"A 25% decrease leaves 75% each year. Multiply the initial $9,000 by 0.75^t.\"},{\"domain\":\"Algebra\",\"prompt\":\"The table shows a linear function g. If g(x)=kx+1, what is k?\",\"options\":[\"\\u22124\",\"2\",\"4\",\"6\"],\"answer\":2,\"drill\":\"Test 9\",\"hint\":\"Divide the change in output by the corresponding change in input.\",\"explanation\":\"Using (0,1) and (3,13), k=(13\\u22121)\\\/(3\\u22120)=4.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test9\\\/question-38.svg\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The table lists 15 travel times. If the smallest and largest are removed, which measure changes the most?\",\"options\":[\"Mode\",\"Median\",\"Range\",\"Mean\"],\"answer\":2,\"drill\":\"Test 9\",\"hint\":\"Compare the old and new minimum and maximum; the middle values stay in place.\",\"explanation\":\"The range falls from 34\\u22125=29 to 22\\u22129=13, a change of 16; the other measures change less.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test9\\\/question-39.svg\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A bar chart summarizes 250 parcels by mass. What is the ratio of parcels in the 6\\u201310 kg group to the 16\\u201320 kg group?\",\"options\":[\"1:2\",\"2:1\",\"3:1\",\"4:1\"],\"answer\":1,\"drill\":\"Test 9\",\"hint\":\"The ratio of counts is the same as the ratio of the displayed percentages.\",\"explanation\":\"The bars show 40% and 20%. Their ratio is 40:20=2:1.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test9\\\/question-40.svg\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A jar contains 12 blue tiles and 20 tiles of other colors. What is the probability a randomly chosen tile is blue?\",\"options\":[\"3\\\/8\",\"3\\\/5\",\"12\\\/20\",\"5\\\/8\"],\"answer\":0,\"drill\":\"Test 9\",\"hint\":\"Divide the number of blue tiles by the total number of tiles.\",\"explanation\":\"There are 32 tiles in total, so P(blue)=12\\\/32=3\\\/8.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"An account starts with $300 and grows by 10% annually. What is the least whole number of years needed to exceed $600?\",\"options\":[\"7\",\"8\",\"9\",\"10\"],\"answer\":1,\"drill\":\"Test 9\",\"hint\":\"Compare 1.10^7 and 1.10^8 with 2.\",\"explanation\":\"Solve 300(1.10)^t\\u003E600. At t=7 the balance is below $600, while at t=8 it is above $600.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The system y=2kx\\u00b2+6x+2 and y\\\/4=\\u2212x has exactly one real solution, where k is positive. What is k?\",\"options\":[\"5\\\/2\",\"5\",\"25\\\/4\",\"25\\\/2\"],\"answer\":2,\"drill\":\"Test 9\",\"hint\":\"Substitute y=\\u22124x and set the discriminant to zero.\",\"explanation\":\"Substitute y=\\u22124x to get 2kx\\u00b2+10x+2=0. Its discriminant is 100\\u221216k, so one solution requires k=25\\\/4.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The graph shows a collectible\\u0027s value y in hundreds of dollars x months after purchase. What does its y-intercept mean?\",\"options\":[\"The purchase value was $4.\",\"The purchase value was $400.\",\"The value rose by $400 in one month.\",\"The value at month 6 was $4,000.\"],\"answer\":1,\"drill\":\"Test 9\",\"hint\":\"The y-intercept gives the value at x = 0 months; account for hundreds of dollars.\",\"explanation\":\"At x=0 the graph shows y=4 hundreds of dollars, so the purchase value was $400.\",\"image\":\"https:\\\/\\\/yourwaytosat.com\\\/wp-content\\\/themes\\\/sat-math-practice-hub-native-v362\\\/public\\\/test9\\\/question-44.svg\"},{\"domain\":\"Algebra\",\"prompt\":\"If 7x + 5 = 47, what is the value of x?\",\"options\":[\"5\",\"6\",\"7\",\"8\"],\"answer\":1,\"drill\":\"Final SAT Math Test\",\"hint\":\"Subtract 5, then divide by 7.\",\"explanation\":\"7x = 42, so x = 6.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The table shows selected values of a quadratic function f. Which expression could define f(x)?\",\"options\":[\"(x \\u2212 1)(x \\u2212 5)\",\"(x + 1)(x + 5)\",\"(x \\u2212 2)(x \\u2212 4)\",\"(x + 2)(x + 4)\"],\"answer\":0,\"drill\":\"Final SAT Math Test\",\"hint\":\"Use the rows where f(x) = 0 to identify the zeros.\",\"explanation\":\"The table shows f(1) = 0 and f(5) = 0, so the factors are x \\u2212 1 and x \\u2212 5. Also, (3 \\u2212 1)(3 \\u2212 5) = \\u22124, matching the table. Therefore f(x) = (x \\u2212 1)(x \\u2212 5).\",\"visual\":\"final-quadratic-table\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A recipe uses 3 cups of flour to make 18 muffins. At this rate, how many cups of flour are needed to make 30 muffins?\",\"options\":[\"4\",\"5\",\"6\",\"7.5\"],\"answer\":1,\"drill\":\"Final SAT Math Test\",\"hint\":\"Use a proportional relationship.\",\"explanation\":\"The rate is 3\\\/18 = 1\\\/6 cup per muffin. For 30 muffins, 30\\\/6 = 5 cups.\"},{\"domain\":\"Algebra\",\"prompt\":\"The graph shows the total charge for renting equipment for different numbers of hours. The relationship is linear. What is the charge for each additional hour?\",\"options\":[\"$7\",\"$12\",\"$19\",\"$26\"],\"answer\":0,\"drill\":\"Final SAT Math Test\",\"hint\":\"Use two labeled points to calculate the change in charge divided by the change in hours.\",\"explanation\":\"From 2 to 5 hours, the charge rises from $26 to $47, an increase of $21 over 3 hours. The hourly charge is $21\\\/3 = $7.\",\"visual\":\"final-service-table\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The graph of y = f(x) is shown. Which equation could define f(x)?\",\"options\":[\"f(x) = (x \\u2212 3)\\u00b2 \\u2212 4\",\"f(x) = (x + 3)\\u00b2 \\u2212 4\",\"f(x) = \\u2212(x \\u2212 3)\\u00b2 \\u2212 4\",\"f(x) = (x \\u2212 3)\\u00b2 + 4\"],\"answer\":0,\"drill\":\"Final SAT Math Test\",\"hint\":\"Identify the vertex and the direction in which the parabola opens.\",\"explanation\":\"The graph opens upward and has vertex (3, \\u22124). In vertex form, this is f(x) = (x \\u2212 3)\\u00b2 \\u2212 4.\",\"visual\":\"final-parabola-graph\"},{\"domain\":\"Algebra\",\"prompt\":\"The graph of line \\u2113 is shown. Which equation represents line \\u2113?\",\"options\":[\"y = 2x + 1\",\"y = 2x \\u2212 1\",\"y = (1\\\/2)x + 1\",\"y = \\u22122x + 1\"],\"answer\":0,\"drill\":\"Final SAT Math Test\",\"hint\":\"Use the two labeled points to find the slope and y-intercept.\",\"explanation\":\"The line passes through (0, 1) and (2, 5). Its slope is (5 \\u2212 1)\\\/(2 \\u2212 0) = 2, and its y-intercept is 1. Therefore y = 2x + 1.\",\"visual\":\"final-line-graph\"},{\"domain\":\"Advanced Math\",\"prompt\":\"If 3^(x+1) = 81, what is the value of x?\",\"options\":[\"2\",\"3\",\"4\",\"5\"],\"answer\":1,\"drill\":\"Final SAT Math Test\",\"hint\":\"Write 81 as a power of 3.\",\"explanation\":\"Since 81 = 3\\u2074, x + 1 = 4 and x = 3.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A jacket originally priced at $120 is discounted by 25%. What is the discounted price?\",\"options\":[\"$30\",\"$80\",\"$90\",\"$95\"],\"answer\":2,\"drill\":\"Final SAT Math Test\",\"hint\":\"The discounted price is 75% of the original price.\",\"explanation\":\"120(0.75) = 90, so the discounted price is $90.\"},{\"domain\":\"Algebra\",\"prompt\":\"Which equation represents a line parallel to y = \\u22124x + 9?\",\"options\":[\"y = 4x + 2\",\"y = \\u22124x \\u2212 3\",\"y = (1\\\/4)x + 9\",\"y = \\u2212(1\\\/4)x \\u2212 3\"],\"answer\":1,\"drill\":\"Final SAT Math Test\",\"hint\":\"Parallel lines have equal slopes.\",\"explanation\":\"The given line has slope \\u22124. Choice B also has slope \\u22124 and a different y-intercept.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"What is the positive solution to x\\u00b2 \\u2212 5x \\u2212 24 = 0?\",\"options\":[\"3\",\"5\",\"8\",\"24\"],\"answer\":2,\"drill\":\"Final SAT Math Test\",\"hint\":\"Factor the quadratic.\",\"explanation\":\"x\\u00b2 \\u2212 5x \\u2212 24 = (x \\u2212 8)(x + 3), so the positive solution is 8.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"A right triangle has legs of lengths 8 and 15. What is the length of its hypotenuse?\",\"options\":[\"16\",\"17\",\"19\",\"23\"],\"answer\":1,\"drill\":\"Final SAT Math Test\",\"hint\":\"Use the Pythagorean theorem.\",\"explanation\":\"The hypotenuse is \\u221a(8\\u00b2 + 15\\u00b2) = \\u221a289 = 17.\"},{\"domain\":\"Algebra\",\"prompt\":\"The system 2x + y = 11 and x \\u2212 y = 1 has solution (x, y). What is x?\",\"options\":[\"3\",\"4\",\"5\",\"6\"],\"answer\":1,\"drill\":\"Final SAT Math Test\",\"hint\":\"Add the equations to eliminate y.\",\"explanation\":\"Adding gives 3x = 12, so x = 4.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The function g is defined by g(x) = 200(0.92)^x. Which statement best describes g?\",\"options\":[\"It decreases by 8% whenever x increases by 1.\",\"It decreases by 92% whenever x increases by 1.\",\"It increases by 8% whenever x increases by 1.\",\"It increases by 92% whenever x increases by 1.\"],\"answer\":0,\"drill\":\"Final SAT Math Test\",\"hint\":\"Compare the multiplier 0.92 with 1.\",\"explanation\":\"A multiplier of 0.92 means 92% remains, which is an 8% decrease per unit increase in x.\"},{\"domain\":\"Algebra\",\"prompt\":\"For what value of k does the equation 3(kx \\u2212 4) = 18x \\u2212 12 have infinitely many solutions?\",\"options\":[\"3\",\"6\",\"9\",\"18\"],\"answer\":1,\"drill\":\"Final SAT Math Test\",\"hint\":\"Expand and match the coefficient of x on both sides.\",\"explanation\":\"The left side is 3kx \\u2212 12. For both sides to be identical, 3k = 18, so k = 6.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"The values 12, 15, 18, 18, and 22 have a mean of 17. If 30 is added to the data set, which measure increases more: the mean or the median?\",\"options\":[\"The mean\",\"The median\",\"They increase by the same amount\",\"Neither increases\"],\"answer\":0,\"drill\":\"Final SAT Math Test\",\"hint\":\"Calculate both new measures.\",\"explanation\":\"The new mean is 115\\\/6 \\u2248 19.17, an increase of about 2.17. The new median is (18 + 18)\\\/2 = 18, unchanged. The mean increases more.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"If x + 2 is a factor of x\\u00b2 + bx \\u2212 12, what is the value of b?\",\"options\":[\"\\u22124\",\"2\",\"4\",\"8\"],\"answer\":0,\"drill\":\"Final SAT Math Test\",\"hint\":\"Substitute x = \\u22122 into the polynomial.\",\"explanation\":\"4 \\u2212 2b \\u2212 12 = 0 gives \\u22122b = 8, so b = \\u22124.\"},{\"domain\":\"Algebra\",\"prompt\":\"A concert venue sold 240 tickets. Adult tickets cost $18 and student tickets cost $12. The total revenue was $3,720. How many student tickets were sold?\",\"options\":[\"80\",\"90\",\"100\",\"120\"],\"answer\":2,\"drill\":\"Final SAT Math Test\",\"hint\":\"Let a + s = 240 and 18a + 12s = 3,720.\",\"explanation\":\"Substitute a = 240 \\u2212 s: 18(240 \\u2212 s) + 12s = 3,720. This gives 4,320 \\u2212 6s = 3,720, so s = 100.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"In right triangle ABC, angle C is 90\\u00b0, AC = 12, and BC = 5. What is sin(A)?\",\"options\":[\"5\\\/13\",\"5\\\/12\",\"12\\\/13\",\"12\\\/5\"],\"answer\":0,\"drill\":\"Final SAT Math Test\",\"hint\":\"Find the hypotenuse, then use opposite over hypotenuse.\",\"explanation\":\"AB = \\u221a(12\\u00b2 + 5\\u00b2) = 13. The side opposite A is BC = 5, so sin(A) = 5\\\/13.\"},{\"domain\":\"Algebra\",\"prompt\":\"Which point (x, y) is a solution to the inequality y \\u003E 2x \\u2212 3?\",\"options\":[\"(0, \\u22124)\",\"(1, \\u22121)\",\"(2, 1)\",\"(3, 4)\"],\"answer\":3,\"drill\":\"Final SAT Math Test\",\"hint\":\"Substitute each point and remember the inequality is strict.\",\"explanation\":\"For (3, 4), 2(3) \\u2212 3 = 3 and 4 \\u003E 3. Choices B and C give equality, not a strict inequality.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"For x \\u2260 5, which expression is equivalent to (x\\u00b2 \\u2212 25)\\\/(x \\u2212 5)?\",\"options\":[\"x \\u2212 5\",\"x + 5\",\"x\\u00b2 + 5\",\"1\"],\"answer\":1,\"drill\":\"Final SAT Math Test\",\"hint\":\"Factor the numerator as a difference of squares.\",\"explanation\":\"x\\u00b2 \\u2212 25 = (x \\u2212 5)(x + 5). Canceling x \\u2212 5 leaves x + 5.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The equation x\\u00b2 \\u2212 6x + c = 0 has exactly one real solution. What is c?\",\"options\":[\"3\",\"6\",\"9\",\"12\"],\"answer\":2,\"drill\":\"Final SAT Math Test\",\"hint\":\"Set the discriminant equal to zero.\",\"explanation\":\"The discriminant is 36 \\u2212 4c. Setting it to 0 gives c = 9.\"},{\"domain\":\"Algebra\",\"prompt\":\"If 4x \\u2212 3y = 19 and y = x \\u2212 2, what is the value of x?\",\"options\":[\"7\",\"9\",\"11\",\"13\"],\"answer\":3,\"drill\":\"Final SAT Math Test\",\"hint\":\"Substitute x \\u2212 2 for y.\",\"explanation\":\"4x \\u2212 3(x \\u2212 2) = 19 gives x + 6 = 19, so x = 13.\"},{\"domain\":\"Algebra\",\"prompt\":\"The function f is defined by f(x) = 5x \\u2212 7. If f(a) = 3a + 9, what is the value of a?\",\"options\":[\"4\",\"6\",\"8\",\"16\"],\"answer\":2,\"drill\":\"Final SAT Math Test\",\"hint\":\"Set the two expressions for f(a) equal.\",\"explanation\":\"5a \\u2212 7 = 3a + 9 gives 2a = 16, so a = 8.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A random sample of 80 students at a school found that 52 support a new schedule. Based on the sample, about how many of the school\\u0027s 1,200 students would be expected to support the schedule?\",\"options\":[\"520\",\"650\",\"720\",\"780\"],\"answer\":3,\"drill\":\"Final SAT Math Test\",\"hint\":\"Use the sample proportion to estimate the population count.\",\"explanation\":\"The sample proportion is 52\\\/80 = 0.65. Then 0.65(1,200) = 780 students.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"The length and width of a rectangle are each increased by 20%. By what percent does the area increase?\",\"options\":[\"20%\",\"40%\",\"44%\",\"144%\"],\"answer\":2,\"drill\":\"Final SAT Math Test\",\"hint\":\"Area is multiplied by the product of the two scale factors.\",\"explanation\":\"The new area factor is 1.2 \\u00d7 1.2 = 1.44, so the area increases by 44%.\"},{\"domain\":\"Algebra\",\"prompt\":\"The graph of 5x \\u2212 2y = 20 intersects the x-axis at (a, 0). What is the value of a?\",\"options\":[\"2\",\"4\",\"5\",\"10\"],\"answer\":1,\"drill\":\"Final SAT Math Test\",\"hint\":\"At an x-intercept, y = 0.\",\"explanation\":\"5a = 20, so a = 4.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"A ball\\u0027s height h, in feet, t seconds after it is thrown is h(t) = \\u221216t\\u00b2 + 64t + 5. After how many seconds does the ball reach its maximum height?\",\"options\":[\"1\",\"2\",\"4\",\"5\"],\"answer\":1,\"drill\":\"Final SAT Math Test\",\"hint\":\"Use the x-coordinate of the vertex, \\u2212b\\\/(2a).\",\"explanation\":\"The vertex occurs at t = \\u221264\\\/(2\\u00b7\\u221216) = 2 seconds.\"},{\"domain\":\"Algebra\",\"prompt\":\"The function f is linear. If f(\\u22122) = 11 and f(4) = \\u22127, which equation defines f?\",\"options\":[\"f(x) = \\u22123x + 5\",\"f(x) = \\u22123x \\u2212 5\",\"f(x) = 3x + 5\",\"f(x) = 3x \\u2212 5\"],\"answer\":0,\"drill\":\"Final SAT Math Test\",\"hint\":\"Find the slope, then use either point to find the intercept.\",\"explanation\":\"The slope is (\\u22127 \\u2212 11)\\\/(4 \\u2212 (\\u22122)) = \\u221218\\\/6 = \\u22123. Using f(\\u22122) = 11 gives 11 = 6 + b, so b = 5.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The solutions to x\\u00b2 \\u2212 10x + 7 = 0 are 5 \\u00b1 \\u221ak. What is the value of k?\",\"options\":[\"7\",\"18\",\"25\",\"32\"],\"answer\":1,\"drill\":\"Final SAT Math Test\",\"hint\":\"Use the quadratic formula and simplify the radical.\",\"explanation\":\"x = (10 \\u00b1 \\u221a(100 \\u2212 28))\\\/2 = (10 \\u00b1 \\u221a72)\\\/2 = 5 \\u00b1 3\\u221a2 = 5 \\u00b1 \\u221a18. Thus k = 18.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"A sector of a circle with radius 9 has a central angle of 80\\u00b0. What is the area of the sector?\",\"options\":[\"9\\u03c0\",\"18\\u03c0\",\"20\\u03c0\",\"36\\u03c0\"],\"answer\":1,\"drill\":\"Final SAT Math Test\",\"hint\":\"Multiply the circle\\u0027s area by 80\\\/360.\",\"explanation\":\"The sector area is (80\\\/360)\\u03c0(9\\u00b2) = (2\\\/9)(81\\u03c0) = 18\\u03c0.\"},{\"domain\":\"Algebra\",\"prompt\":\"For which value of c does the system y = 3x + 4 and 6x \\u2212 2y = c have infinitely many solutions?\",\"options\":[\"\\u22128\",\"0\",\"4\",\"8\"],\"answer\":0,\"drill\":\"Final SAT Math Test\",\"hint\":\"Substitute the first equation into the second.\",\"explanation\":\"Substitution gives 6x \\u2212 2(3x + 4) = \\u22128 for every x. Therefore the two equations represent the same line when c = \\u22128, giving infinitely many solutions.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"A study randomly assigned 200 volunteers to use either a new study method or the usual method. The new-method group earned significantly higher scores. For which population can the study\\u0027s causal conclusion be made?\",\"options\":[\"All students everywhere\",\"Only the 200 volunteers\",\"All students at the volunteers\\u0027 schools\",\"Any student who chooses the new method\"],\"answer\":1,\"drill\":\"Final SAT Math Test\",\"hint\":\"Random assignment supports causation, but volunteers are not a random sample of a larger population.\",\"explanation\":\"Random assignment supports a causal conclusion for the participants. Because the volunteers were not randomly sampled from a larger population, the conclusion cannot automatically be generalized beyond them.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"If \\u221a(3x + 4) = x, what is the positive value of x?\",\"options\":[\"1\",\"4\",\"5\",\"8\"],\"answer\":1,\"drill\":\"Final SAT Math Test\",\"hint\":\"Square both sides, then check the solutions in the original equation.\",\"explanation\":\"Squaring gives 3x + 4 = x\\u00b2, or x\\u00b2 \\u2212 3x \\u2212 4 = 0 = (x \\u2212 4)(x + 1). The positive solution is 4, and it checks in the original equation.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"Two similar triangles have corresponding side lengths 6 and 15. If the area of the smaller triangle is 24, what is the area of the larger triangle?\",\"options\":[\"60\",\"96\",\"120\",\"150\"],\"answer\":3,\"drill\":\"Final SAT Math Test\",\"hint\":\"Areas scale by the square of the side-length scale factor.\",\"explanation\":\"The side scale factor is 15\\\/6 = 5\\\/2. The area factor is (5\\\/2)\\u00b2 = 25\\\/4, so the larger area is 24(25\\\/4) = 150.\"},{\"domain\":\"Algebra\",\"prompt\":\"A company charges a one-time setup fee plus a constant monthly fee. After 4 months, the total is $146. After 10 months, the total is $284. What is the setup fee?\",\"options\":[\"$23\",\"$54\",\"$69\",\"$92\"],\"answer\":1,\"drill\":\"Final SAT Math Test\",\"hint\":\"Find the monthly rate from the change in total cost.\",\"explanation\":\"The monthly rate is (284 \\u2212 146)\\\/(10 \\u2212 4) = 138\\\/6 = 23. The setup fee is 146 \\u2212 4(23) = 54 dollars.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"For positive x, the expression (16x^12)^(1\\\/4) is equivalent to which expression?\",\"options\":[\"2x\\u00b3\",\"4x\\u00b3\",\"8x\\u2074\",\"4x\\u2078\"],\"answer\":0,\"drill\":\"Final SAT Math Test\",\"hint\":\"Apply the fourth root to the coefficient and the exponent.\",\"explanation\":\"The fourth root of 16 is 2, and (x^12)^(1\\\/4) = x\\u00b3 for positive x. The expression is 2x\\u00b3.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"In a group of 120 students, 70 study French, 50 study Spanish, and 30 study both. If one student who studies French is selected at random, what is the probability that the student also studies Spanish?\",\"options\":[\"1\\\/4\",\"3\\\/7\",\"1\\\/2\",\"3\\\/5\"],\"answer\":1,\"drill\":\"Final SAT Math Test\",\"hint\":\"Use the number who study both divided by the number who study French.\",\"explanation\":\"Among the 70 French students, 30 also study Spanish. The conditional probability is 30\\\/70 = 3\\\/7.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The graph of y = a(x \\u2212 2)(x + 6) passes through (0, \\u221224). What is the value of a?\",\"options\":[\"\\u22122\",\"\\u22121\\\/2\",\"1\\\/2\",\"2\"],\"answer\":3,\"drill\":\"Final SAT Math Test\",\"hint\":\"Substitute x = 0 and y = \\u221224.\",\"explanation\":\"\\u221224 = a(\\u22122)(6) = \\u221212a, so a = 2.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"In circle O, a tangent at point T and radius OT form an angle of 90\\u00b0. Point P lies outside the circle, PT = 12, and OP = 13. What is the radius of the circle?\",\"options\":[\"5\",\"7\",\"12\",\"25\"],\"answer\":0,\"drill\":\"Final SAT Math Test\",\"hint\":\"Triangle OTP is right at T.\",\"explanation\":\"OT\\u00b2 + PT\\u00b2 = OP\\u00b2, so r\\u00b2 + 12\\u00b2 = 13\\u00b2. Thus r\\u00b2 = 25 and r = 5.\"},{\"domain\":\"Algebra\",\"prompt\":\"If 2 \\u003C 3x \\u2212 4 \\u2264 17, which interval contains all possible values of x?\",\"options\":[\"2 \\u003C x \\u2264 7\",\"\\u22122 \\u003C x \\u2264 7\",\"2 \\u2264 x \\u003C 7\",\"6 \\u003C x \\u2264 21\"],\"answer\":0,\"drill\":\"Final SAT Math Test\",\"hint\":\"Add 4 to all three parts, then divide by 3.\",\"explanation\":\"Adding 4 gives 6 \\u003C 3x \\u2264 21. Dividing by 3 gives 2 \\u003C x \\u2264 7.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"A polynomial p(x) has zeros \\u22123, 2, and 2. Which expression could define p(x)?\",\"options\":[\"(x \\u2212 3)(x + 2)\\u00b2\",\"(x + 3)(x \\u2212 2)\\u00b2\",\"(x + 3)\\u00b2(x \\u2212 2)\",\"(x \\u2212 3)\\u00b2(x + 2)\"],\"answer\":1,\"drill\":\"Final SAT Math Test\",\"hint\":\"A zero r corresponds to a factor x \\u2212 r; repeated zeros repeat the factor.\",\"explanation\":\"The zero \\u22123 gives factor x + 3, and the repeated zero 2 gives (x \\u2212 2)\\u00b2. Thus choice B works.\"},{\"domain\":\"Problem-Solving and Data Analysis\",\"prompt\":\"Two random samples estimate the proportion of voters supporting a proposal. Sample A has 100 voters and Sample B has 1,600 voters. Both report 52% support. Which sample is expected to have the smaller margin of error?\",\"options\":[\"Sample A\",\"Sample B\",\"The margins must be equal\",\"There is not enough information\"],\"answer\":1,\"drill\":\"Final SAT Math Test\",\"hint\":\"Larger random samples generally produce smaller margins of error.\",\"explanation\":\"Sample B is much larger, so its estimate is expected to have a smaller margin of error.\"},{\"domain\":\"Geometry and Trigonometry\",\"prompt\":\"The equation x\\u00b2 + y\\u00b2 \\u2212 8x + 6y = 0 represents a circle. What is the radius of the circle?\",\"options\":[\"3\",\"4\",\"5\",\"7\"],\"answer\":2,\"drill\":\"Final SAT Math Test\",\"hint\":\"Complete the square in x and y.\",\"explanation\":\"Rewriting gives (x \\u2212 4)\\u00b2 \\u2212 16 + (y + 3)\\u00b2 \\u2212 9 = 0, so (x \\u2212 4)\\u00b2 + (y + 3)\\u00b2 = 25. The radius is 5.\"},{\"domain\":\"Advanced Math\",\"prompt\":\"The function f(x) = 3(1.5)^x. 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