The S A T® - SAT Suite of Assessments



The S?A?T?Assistive Technology Compatible Test FormPractice Test 1Answers and explanations for section?3, Math Test—No CalculatorExplanation for question 1.Correct answerChoice?D is correct. Since k equals 3, one can substitute 3 for k in the equation the fraction with numerator x minus 1, and denominator 3, equals k, which gives the fraction with numerator x minus 1, and denominator 3, equals 3. Multiplying both sides of the equation, the fraction with numerator x minus 1, and denominator 3, equals 3 by 3 gives x minus 1, equals 9 and then adding 1 to both sides of x minus 1, equals?9 gives x?equals?10.Incorrect answerChoices?A, B, and C are incorrect because the result of subtracting 1 from the value and dividing by 3 is not the given value of k, which is?3.Explanation for question 2.Correct answerChoice?A is correct. To calculate open parenthesis, 7 plus 3?i, close parenthesis, plus, open parenthesis, negative?8 plus 9?i, close parenthesis, add the real parts of each complex number, 7 plus, negative?8, equals negative?1, and then add the imaginary parts, 3?i, plus 9?i, equals 12?i. The result is negative?1 plus?12?i.Incorrect answerChoices?B, C, and D are incorrect and likely result from common errors that arise when adding complex numbers. For example, choice?B is the result of adding 3?i and negative?9?i, and choice?C is the result of adding 7 and?8.Explanation for question 3.Correct answerChoice?C is correct. The total number of text messages sent by Armand can be found by multiplying his rate of texting, in number of text messages sent per hour, by the total number of hours he spent sending them; that is m?texts per hour, times 5?hours, equals 5?m texts. Similarly, the total number of text messages sent by Tyrone is his hourly rate of texting multiplied by the 4?hours he spent texting: p?texts per hour, times 4?hours, equals 4?p. The total number of text messages sent by Armand and Tyrone is the sum of the total number of messages sent by Armand and the total number of messages sent by Tyrone: 5?m?plus?4?p.Incorrect answerChoice?A is incorrect and arises from adding the coefficients and multiplying the variables of 5?m and 4?p. Choice?B is incorrect and is the result of multiplying 5?m and 4?p. The total number of messages sent by Armand and Tyrone should be the sum of 5?m and 4?p, not the product of these terms. Choice?D is incorrect because it multiplies Armand’s number of hours spent texting by Tyrone’s hourly rate of texting, and vice versa. This mixup results in an expression that does not equal the total number of messages sent by Armand and Tyrone.Explanation for question 4.Correct answerChoice?B is correct. The value 108 in the equation is the value of P in P equals, 108 minus 23?d when d equals 0. When d equals 0, Kathy has worked 0?days that week. In other words, 108 is the number of phones left before Kathy has started work for the week. Therefore, the meaning of the value 108 in the equation is that Kathy starts each week with 108?phones to fix.Incorrect answerChoice?A is incorrect because Kathy will complete the repairs when P?equals 0. Since P equals, 108 minus 23?d, this will occur when 0 equals 108 minus 23?d or when d equals, 108 over 23, not when d equals 108. Therefore, the value 108 in the equation does not represent the number of days it will take Kathy to complete the repairs. Choices?C and D are incorrect because the number?23 in P equals, 108 minus?23?d indicates that the number of phones left will decrease by 23 for each increase in the value of d by 1; in other words, Kathy is repairing phones at a rate of 23?per day, not 108 per hour (choice?C) or 108?per day (choice?D).Explanation for question 5.Correct answerChoice?C is correct. Only like terms, with the same variables and exponents, can be combined to determine the answer as shown here: open parenthesis, x?squared y, minus 3?y?squared, plus, 5?x, y?squared, close parenthesis, minus, open parenthesis, negative x?squared y, plus, 3?x, y?squared, minus 3?y?squared, close parenthesis equals, open parenthesis, x?squared y, minus, open parenthesis, negative x?squared y, close parenthesis, close?parenthesis, plus, open parenthesis, negative?3, y?squared, minus, open?parenthesis, negative?3, y?squared, close parenthesis, close parenthesis, plus, open parenthesis, 5?x, y?squared, minus 3?x, y?squared, close parenthesis which equals, 2?x?squared y, plus 0, plus 2?x, y?squared which equals, 2?x?squared y, plus 2?x, y?squaredIncorrect answerChoices?A, B, and D are incorrect and are the result of common calculation errors or of incorrectly combining like and unlike terms.Explanation for question 6.Correct answerChoice?A is correct. In the equation h equals, 3?a, plus 28.6, if a, the age of the boy, increases by 1, then h becomes h equals, 3 times, open?parenthesis, a, plus 1, close parenthesis, plus, 28.6, equals, 3?a, plus 3, plus?28.6, which equals, open parenthesis, 3?a, plus 28.6, close parenthesis, plus?3. Therefore, the model estimates that the boy’s height increases by 3?inches each?year.Alternatively: The height, h, is a linear function of the age, a, of the boy. The coefficient 3 can be interpreted as the rate of change of the function; in this case, the rate of change can be described as a change of 3?inches in height for every additional year in age.Incorrect answerChoices?B, C, and D are incorrect and are likely the result of dividing 28.6 by 5, 3, and 2, respectively. The number 28.6 is the estimated height, in inches, of a newborn boy. However, dividing 28.6 by 5, 3, or 2 has no meaning in the context of this question.Explanation for question 7.Correct answerChoice?B is correct. Since the righthand side of the equation is P times the expression the fraction with numerator open parenthesis, r?over 1,200, close parenthesis, times, open parenthesis, 1 plus, the fraction r over 1,200, close parenthesis, raised to the N power, and denominator open parenthesis, 1 plus, the fraction r over 1,200, close parenthesis, raised to the N power, minus 1, end fraction, multiplying both sides of the equation by the reciprocal of this expression results in the fraction with numerator open parenthesis, 1 plus, the fraction r over 1,200, close parenthesis, raised to the N power, minus 1, and denominator open parenthesis, r over 1,200, close?parenthesis, times, open parenthesis, 1 plus, the fraction r over 1,200, close?parenthesis, raised to the N power, end fraction, times m, equals?P.Incorrect answerChoice?A is incorrect and is the result of multiplying both sides of the equation by the rational expression the fraction with numerator open?parenthesis, r over 1,200, close parenthesis, times, open parenthesis, 1 plus, the fraction r over 1,200, close parenthesis, raised to the N power, and denominator open parenthesis, 1 plus, the fraction r over 1,200, close parenthesis, raised to the N power, minus 1, end fraction rather than by the reciprocal of this expression the fraction with numerator open parenthesis, 1 plus, the fraction r over 1,200, close parenthesis, raised to the N power, minus 1, and denominator open parenthesis, r over 1,200, close parenthesis, times, open?parenthesis, 1 plus, the fraction r over 1,200, close parenthesis, raised to the N power, end fraction. Choices?C and D are incorrect and are likely the result of errors while trying to solve for?P.Explanation for question 8.Correct answerChoice?C is correct. Since a, over b equals 2, it follows that b over a, equals 1 over 2. Multiplying both sides of the equation by 4 gives 4 times, b over a, equals 4 times, 1 over 2, or 4?b, over a, equals 2.Incorrect answerChoice A is incorrect because if 4?b, over a, equals 0, then a over b would be undefined. Choice?B is incorrect because if 4?b, over a, equals 1, then a over b, equals 4. Choice?D is incorrect because if 4?b, over a, equals 4, then a, over b, equals?1.Explanation for question 9.Correct answerChoice?B is correct. Adding x and 19 to both sides of 2?y minus x, equals negative?19 gives x equals, 2?y plus 19. Then, substituting 2?y plus 19 for x in 3?x plus 4?y equals negative?23 gives 3?times, open parenthesis, 2?y plus 19, close parenthesis, plus 4?y, equals negative?23. This last equation is equivalent to 10?y, plus 57, equals negative?23. Solving 10?y, plus 57, equals negative?23 gives y equals, negative?8. Finally, substituting negative?8 for y in 2?y minus x, equals negative?19 gives 2?times, open parenthesis, negative?8, close parenthesis, minus x, equals negative?19, or x equals 3. Therefore, the solution the ordered pair x?comma y to the given system of equations is the ordered pair 3 comma negative?8.Incorrect answerChoices?A, C, and D are incorrect because when the given values of x and y are substituted in 2?y minus x, equals negative?19, the value of the left side of the equation does not equal negative?19.Explanation for question 10.Correct answerChoice?A is correct. Since g is an even function, g of negative?4, equals g of 4, which equals?8.Alternatively: First find the value of a, and then find g of negative?4. Since g of 4, equals 8, substituting 4 for x and 8 for g of x gives 8 equals, a, times, open parenthesis, 4, close parenthesis, squared, plus 24, equals, 16?a, plus 24. Solving this last equation gives a,?equals negative?1. Thus g of x, equals, negative, x?squared, plus 24, from which it follows that g of negative 4, equals negative, open parenthesis, negative?4, close parenthesis, squared, plus 24; g of negative?4 equals negative?16 plus 24; and g of negative?4 equals?8.Incorrect answerChoices?B, C, and D are incorrect because g is a function and there can only be one value of g of negative?4.Explanation for question 11.Correct answerChoice?D is correct. To determine the price per pound of beef when it was equal to the price per pound of chicken, determine the value of x (the number of weeks after July?1) when the two prices were equal. The prices were equal when b?equals c; that is, when 2.35, plus 0.25?x, equals, 1.75, plus 0.40?x. This last equation is equivalent to 0.60 equals, 0.15?x, and so x equals, 0.60, over 0.15, which equals 4. Then to determine b, the price per pound of beef, substitute 4 for x in b?equals, 2.35 plus 0.25?x, which gives b equals, 2.35 plus, 0.25 times 4, which equals 3.35 dollars per pound.Incorrect answerChoice?A is incorrect. It results from substituting the value?1, not 4, for x in b equals, 2.35, plus 0.25?x. Choice?B is incorrect. It results from substituting the value?2, not 4, for x in b equals, 2.35, plus 0.25?x. Choice?C is incorrect. It results from substituting the value?3, not 4, for x in c equals, 1.75, plus?0.40?x.Explanation for question 12.Correct answerChoice?D is correct. In the x?yplane, all lines that pass through the origin are of the form y equals, m?x, where m is the slope of the line. Therefore, the equation of this line is y equals, one seventh?x, or x equals 7?y. A?point with coordinates a, comma b will lie on the line if and only if a, equals 7?b. Of the given choices, only choice?D, the point with coordinates 14 comma 2, satisfies this condition: 14 equals 7 times?2.Incorrect answerChoice?A is incorrect because the line determined by the origin with?coordinates 0 comma 0 and the point with coordinates 0 comma 7 is the vertical line with equation x equals 0; that is, the yaxis. The slope of the yaxis is undefined, not one seventh. Therefore, the point with?coordinates 0 comma 7 does not lie on the line that passes the origin and has slope one?seventh. Choices?B and C are incorrect because neither of the ordered pairs has a ycoordinate that is one seventh the value of the corresponding xcoordinate.Explanation for question 13.Correct answerChoice?B is correct. To rewrite the complex fraction with numerator 1, and denominator the fraction 1 over x plus 2, end fraction, plus, the fraction 1?over x plus 3, end fraction, end complex fraction, multiply by the?fraction with numerator open parenthesis, x plus 2, close parenthesis, times, open parenthesis, x plus 3, close parenthesis, and denominator open parenthesis, x?plus 2, close parenthesis, times, open parenthesis, x plus 3, close parenthesis, end fraction. This results in the expression the fraction with numerator open parenthesis, x plus 2, close parenthesis, times, open parenthesis, x?plus 3, close parenthesis, and denominator open parenthesis, x plus 2, close parenthesis, plus, open parenthesis, x plus 3, close parenthesis, end fraction, which is equivalent to the expression in choice?B.Incorrect answerChoices?A, C, and D are incorrect and could be the result of common algebraic errors that arise while manipulating a complex fraction.Explanation for question 14.Correct answerChoice?A is correct. One approach is to express the fraction with numerator 8 raised to the x power, and denominator 2 raised to the y power, end fraction so that the numerator and denominator are expressed with the same base. Since 2 and 8 are both powers of 2, substituting 2?cubed for 8 in the numerator of the fraction with numerator 8 raised to the x power, and denominator 2 raised to the y?power, end fraction gives the fraction with numerator open parenthesis, 2?cubed, close parenthesis, raised to the x power, and denominator 2 raised to the y?power, end fraction, which can be rewritten as the fraction with numerator 2 raised to the 3?x power, and denominator 2 raised to the y?power, end fraction. Since the numerator and denominator of the fraction with numerator 2 raised to the 3?x power, and denominator 2 raised to the y?power, end fraction have a common base, this expression can be rewritten as 2 raised to the 3?x minus y?power. It is given that 3?x minus y, equals 12, so one can substitute 12 for the exponent, 3?x minus y, given that the expression the fraction with numerator 8 raised to the x power, and denominator 2 raised to the y?power, end fraction is equal to 2 raised to the twelfth power.Incorrect answerChoice?B is incorrect. The expression the fraction with numerator 8 raised to the x power, and denominator 2 raised to the y power, end fraction can be rewritten as the fraction with numerator 2 raised to the 3?x power, and denominator 2 raised to the y?power, end fraction, or 2 raised to the 3?x minus y?power. If the value of 2 raised to the 3?x minus y?power is 4 to the fourth power, which can be rewritten as 28, then 2 raised to the 3?x minus y?power, equals 2 to the eighth power, which results in 3?x minus y, equals 8 not 12. Choice?C is incorrect. If the value of the fraction with numerator 8 raised to the x power, and denominator 2 raised to the y power, end fraction is 8?squared, then 2 raised to the 3?x minus y?power, equals 2 to the eighth power, which results in 3?x minus y, equals 6, not 12. Choice?D is incorrect because the value of the fraction with numerator 8 raised to the x power, and denominator 2 raised to the y power, end fraction can be determined.Explanation for question 15.Correct answerChoice?D is correct. One can find the possible values of a and b in open parenthesis, a, x plus 2, close parenthesis, times, open parenthesis, b?x plus 7, close parenthesis by using the given equation a, plus b, equals 8 and finding another equation that relates the variables a and b. Since open parenthesis, a, x plus 2, close parenthesis, times, open parenthesis, b?x plus 7, close parenthesis, equals, 15?x?squared, plus c?x, plus 14, one can expand the left side of the equation to obtain a, b?x?squared, plus 7?a, x, plus 2?b?x, plus 14, equals, 15?x?squared, plus c?x, plus 14. Since a?b is the coefficient of x?squared on the left side of the equation and 15 is the coefficient of x?squared on the right side of the equation, it must be true that a, b equals 15. Since a, plus b, equals 8, it follows that b equals, 8 minus a. Thus, a, b equals 15 can be rewritten as a, times, open parenthesis, 8 minus a, close parenthesis, equals 15, which in turn can be rewritten as a, squared, minus 8?a, plus 15, equals 0. Factoring gives open parenthesis, a, minus 3, close parenthesis, times, open parenthesis, a, minus 5, close parenthesis, equals 0. Thus, either a, equals 3 and b equals 5, or a, equals 5 and b equals 3. If a, equals 3 and b equals 5, then open parenthesis, a, x plus 2, close parenthesis, times, open parenthesis, b?x plus 7, close parenthesis, equals, open parenthesis, 3?x plus 2, close parenthesis, times, open parenthesis, 5?x plus 7, close parenthesis, which equals 15?x?squared, plus 31?x, plus 14. Thus, one of the possible values of c is 31. If a, equals 5 and b?equals 3, then open parenthesis, a, x plus 2, close parenthesis, times, open parenthesis, b?x plus 7, close parenthesis, equals, open parenthesis, 5?x plus 2, close parenthesis, times, open parenthesis, 3?x plus 7, close parenthesis, which equals 15?x?squared, plus 41?x, plus?14. Thus, another possible value for c is 41. Therefore, the two possible values for c are 31 and?41.Incorrect answerChoice?A is incorrect; the numbers 3 and 5 are possible values for a and b, but not possible values for c. Choice?B is incorrect; if a, equals 5 and b?equals 3, then 6 and 35 are the coefficients of x when the expression open parenthesis, 5?x plus 2, close parenthesis, times, open parenthesis, 3?x plus 7, close parenthesis is expanded as 15?x?squared, plus 35?x, plus 6?x, plus 14. However, when the coefficients of x are 6 and 35, the value of c is 41 and not 6 and 35. Choice?C is incorrect; if a,?equals 3, equals 3 and b equals 5, then 10 and 21 are the coefficients of x when the expression open parenthesis, 3?x plus 2, close parenthesis, times, open parenthesis, 5?x plus 7, close parenthesis is expanded as 15?x?squared, plus 21?x, plus 10?x, plus 14. However, when the coefficients of x are 10 and 21, the value of c is 31 and not 10 and?21.Explanation for question 16.Correct answerThe correct answer is 2. To solve for t, factor the left side of t?squared, minus 4, equals 0, giving open parenthesis, t minus 2, close parenthesis, times, open parenthesis, t plus 2, close parenthesis, equals 0. Therefore, either t minus 2, equals 0 or t plus 2, equals 0. If t minus 2, equals 0, then t equals 2, and if t plus 2, equals?0, then t equals negative?2. Since it is given that t is greater than 0, the value of t must be?2.Another way to solve for t is to add 4 to both sides of t?squared, minus 4, equals 0, giving t?squared equals 4. Then, taking the square root of the left and the right side of the equation gives t equals, plus or minus the square root of 4, which equals plus or minus 2. Since it is given that t is greater than 0, the value of t must be?2.Explanation for question 17.Correct answerThe correct answer is 1600. It is given that angle?A?E?B and angle?C?D?B have the same measure. Since angle?A?B?E and angle?C?D?B are vertical angles, they have the same measure. Therefore, triangle?E?A?B is similar to triangle?D?C?B because the triangles have two pairs of congruent corresponding angles (angleangle criterion for similarity of triangles). Since the triangles are similar, the corresponding sides are in the same proportion; thus the length of side?C?D over x, equals, the length of side?B?D over the length of side?E?B. Substituting the given values of 800 for the length of C?D, 700 for the length of B?D, and 1400 for the length of E?B in the length of side?C?D over x, equals, the length of side?B?D over the length of side?E?B gives 800 over x, equals, 700 over 1,400. Therefore, x equals, the fraction with numerator 800 times 1,400, and denominator 700, which equals?1,600.Explanation for question 18.Correct answerThe correct answer is 7. Subtracting the left and right sides of x plus y equals, negative?9 from the corresponding sides of x plus 2?y equals, negative?25 gives open parenthesis, x plus 2?y, close parenthesis, minus, open parenthesis, x plus y, close parenthesis, equals, negative?25, minus negative?9, which is equivalent to y equals negative?16. Substituting negative?16 for y in x plus y equals, negative?9 gives x plus negative?16, equals negative?9, which is equivalent to x equals, negative?9 minus negative?16, which equals?7.Explanation for question 19.Correct answerThe correct answer is 4 over 5 or 0.8. By the complementary angle relationship for sine and cosine, sine of x?degrees, equals, cosine of, open parenthesis, 90?degrees minus x?degrees, close parenthesis. Therefore, cosine of, open parenthesis, 90?degrees minus x?degrees, close parenthesis, equals 4 over 5. Either the fraction 4 over 5 or its decimal equivalent, 0.8, may be gridded as the correct answer.Alternatively, one can construct a right triangle that has an angle of measure ?x?degrees such that sine of x?degrees, equals, 4 over 5, as shown in the following figure, where sine of x?degrees is equal to the ratio of the length of the side opposite the angle measuring x?degrees to the length of the hypotenuse, or 4?over?5.Begin skippable figure description.The figure presents a right triangle. The vertical side is labeled 4, the hypotenuse is labeled 5, and the horizontal side is not labeled. The angle opposite the vertical side is labeled “x?degrees,” the angle opposite the hypotenuse is labeled “90?degrees,” and the angle opposite the horizontal side is labeled “90?degrees minus x?degrees.”End skippable figure description.Since two of the angles of the triangle are of measure x?degrees and ?90?degrees, the third angle must have the measure 180?degrees, minus 90?degrees, minus x?degrees, equals, 90?degrees minus x?degrees. From the figure, cosine of, open parenthesis, 90?degrees minus x?degrees, close parenthesis, which is equal to the ratio of the length of the side adjacent to the angle measuring 90?degrees minus x?degrees to the hypotenuse, is also 4?over?5.Explanation for question 20.Correct answerThe correct answer is 100. Since a, equals 5 times the square root of 2, one can substitute 5 times the square root of 2 for a in 2?a, equals the square root of 2?x, end root, giving 10 times the square root of 2, equals, the square root of 2?x, end root. Squaring each side of 10?times the square root of 2, equals, the square root of 2?x, end root gives open parenthesis, 10 times the square root of 2, close parenthesis, squared, equals, open parenthesis, the square root of 2?x, end root, close parenthesis, squared, which simplifies to 10?squared, times, open parenthesis, the square root of 2, close parenthesis, squared, equals, open parenthesis, the square root of 2?x, end root, close parenthesis, squared, or 200 equals 2?x. This gives x equals 100. To verify, substitute 100 for x and 5 times the square root of 2 for a in the equation 2?a, equals, the square root of 2?x, end root, which yields 2?times, open parenthesis, 5 times the square root of 2, close parenthesis, equals, the square root of 2 times 100, end root; this is true since 2 times, open parenthesis, 5 times the square root of 2, close parenthesis, equals, 10 times the square root of 2 and the square root of 2 times 100, end root, equals, the square root of 2, end root, times the square root of 100, equals 10 times the square root of?2. ................
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