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The S A T®

Assistive Technology Compatible Test Form

Practice Test 3Answers and explanations for section 3, Math Test—No Calculator

Explanation for question 1.Correct answerChoice C is correct. The painter’s fee is given by n K l h, where n is the number of walls, K is a constant with units of dollars per square foot, l is the length of each wall in feet, and h is the height of each wall in feet. Examining this equation shows that l and h will be used to determine the area of each wall. The variable n is the number of walls, so n times the area of each wall will give the amount of area that will need to be painted. The only remaining variable is K, which represents the cost per square foot and is determined by the painter’s time and the price of paint. Therefore, K is the only factor that will change if the customer asks for a more expensive brand of paint.Incorrect answerChoice A is incorrect because a more expensive brand of paint would not cause the height of each wall to change. Choice B is incorrect because a more expensive brand of paint would not cause the length of each wall to change. Choice D is incorrect because a more expensive brand of paint would not cause the number of walls to change.

Explanation for question 2.Correct answerChoice D is correct. Dividing each side of the equation 3 r equals 18 by 3 gives r equals 6. Substituting 6 for r in the expression 6 r plus 3

gives 6 times 6, plus 3, equals 39.

Alternatively, the expression 6 r plus 3 can be rewritten as

2 times 3 r, plus 3. Substituting 18 for 3 r in the expression 2 times 3 r,

plus 3 yields 2 times 18, plus 3, or 36 plus 3, equals 39.Incorrect answerChoice A is incorrect because 6 is the value of r; however, the question asks for the value of the expression 6 r plus 3. Choices B and C are incorrect because if 6 r plus 3 were equal to either of these values, then it would not be possible for 3 r to be equal to 18, as stated in the question.

Explanation for question 3.Correct answer

Choice D is correct. By definition, a, raised to the m over n power, equals the n root of, a, to the m power, end root for any positive integers m and n.

It follows, therefore, that a, to the two thirds power, equals the cube root of, a, squared, end root.Incorrect answer

Choice A is incorrect. By definition, a, raised to the 1 over n power, equals the n root of a for any positive integer n. Applying this definition as well as

the power property of exponents to the expression the square root of, a, to

the one third power, end root yields the square root of, a, to the one third power, end root, equals open parenthesis, a, to the one third power, close parenthesis, raised to the one half power, which equals a, to the one sixth power.

Because a, to the one sixth power is not equals to a, to the two thirds

power, the square root of, a, to the one third power, end root is not the

correct answer. Choice B is incorrect. By definition, a, raised to the 1 over n power, equals the n root of a for any positive integer n. Applying this

definition as well as the power property of exponents to the expression the

square root of, a, cubed, end root yields the square root of, a, cubed, end root, equals open parenthesis, a, cubed, close parenthesis, raised to the

one half power, which equals a, to the three halves power. Because a, to

the three halves power, is not equal to a, to the two thirds power, the square root of, a, cubed, end root is not the correct answer. Choice C is incorrect. By

definition, a, raised to the 1 over n power, equals the n root of a for any positive integer n. Applying this definition as well as the power property of

exponents to the expression the cube root of, a, to the one half power,

end root yields the cube root of, a, to the one half power, end root, equals open parenthesis, a, to the one half power, close parenthesis, raised to

the one third power, which equals a, to the one sixth power. Because a, to

the one sixth power, is not equals to a, to the two thirds power, the cube root of, a, to the one half power, end root is not the correct answer.

Explanation for question 4.Correct answerChoice B is correct. To fit the scenario described, 30 must be twice as large as x. This can be written as 2 x equals 30.Incorrect answerChoices A, C, and D are incorrect. These equations do not correctly relate the numbers and variables described in the stem. For example, the expression in choice C states that 30 is half as large as x, not twice as large as x.


Choice C is correct. Multiplying each side of the fraction 5 over x, equals the fraction with numerator 15, and denominator x plus 20, end fraction by

x times, open parenthesis, x plus 20, close parenthesis gives

5 times, open parenthesis, x plus 20, close parenthesis, equals 15 x. Using the distributive property to eliminate the parentheses yields 5 x plus 100, equals 15 x, and then subtracting 5 x from each side of the equation

5 x plus 100, equals 15 x gives 100 equals 10 x. Finally, dividing both sides of the equation 100 equals 10 x by 10 gives

10 equals x. Therefore, the value of x over 5 is 10 over 5, equals 2.Incorrect answer

Choice A is incorrect because it is the value of x, not x over 5. Choices B and D are incorrect and may be the result of errors in arithmetic operations on the given equation.

Explanation for question 6.Correct answerChoice C is correct. Multiplying each side of the equation 2 x minus 3 y, equals negative 14 by 3 gives 6 x minus 9 y, equals negative 42. Multiplying each side of the equation 3 x minus 2 y, equals negative 6 by 2 gives 6 x minus 4 y, equals negative 12. Then, subtracting the sides of 6 x minus 4 y, equals negative 12 from the corresponding sides of 6 x minus 9 y, equals negative 42 gives negative 5 y equals negative 30. Dividing each side of the equation negative 5 y equals negative 30 by negative 5 gives y equals 6. Finally, substituting 6 for y in 2 x minus 3 y, equals negative 14 gives

2 x minus, 3 times 6, equals negative 14, or x equals 2. Therefore, the value of x minus y is 2 minus 6, equals negative 4.

Alternatively, adding the corresponding sides of 2 x minus 3 y equals negative 14 and 3 x minus 2 y, equals negative 6 gives

5 x minus 5 y, equals negative 20, from which it follows that x minus y, equals negative 4.

Incorrect answerChoices A and B are incorrect and may be the result of an arithmetic error when solving the system of equations. Choice D is incorrect and may be the result of finding x plus y instead of x minus y.


Choice C is correct. If x minus b is a factor of f of x, then

f of b must equal 0. Based on the table, f of 4 equals 0. Therefore,

x minus 4 must be a factor of f of x.Incorrect answer

Choice A is incorrect because f of 2 is not equal to 0. Choice B is

incorrect because no information is given about the value of f of 3, so

x minus 3 may or may not be a factor of f of x. Choice D is incorrect

because f of 5 is not equal to 0.

Explanation for question 8.Correct answerChoice A is correct. The linear equation y equals, k x plus 4 is in slope-intercept form, and so the slope of the line is k. Since the line contains the

point with coordinates c, comma d, the coordinates of this point satisfy the equation y equals, k x plus 4; therefore, d equals, k c plus 4.

Solving this equation for the slope, k, gives k equals, the fraction with numerator d minus 4, and denominator c.Incorrect answerChoices B, C, and D are incorrect and may be the result of errors in substituting the

coordinates of c, comma d in y equals, k x plus 4 or of errors in solving for k in the resulting equation.

Explanation for question 9.Correct answerChoice A is correct. If a system of two linear equations has no solution, then the lines represented by the equations in the coordinate plane are parallel. The equation

k x minus 3 y, equals 4 can be rewritten as y equals, the

fraction k over 3, end fraction, times x, minus four thirds, where the fraction k over 3 is the slope of the line, and the equation 4 x minus 5 y,

equals 7 can be rewritten as y equals, four fifths x, minus seven fifths,

where four fifths is the slope of the line. If two lines are parallel, then the slopes

of the line are equal. Therefore, four fifths equals the fraction k over 3, or

k equals twelve fifths. (Since the y-intercepts of the lines represented by the

equations are negative four thirds and negative seven fifths, the lines are parallel, not identical.) Incorrect answerChoices B, C, and D are incorrect and may be the result of a computational error when rewriting the equations or solving the equation representing the equality of the slopes for k.


Choice A is correct. Substituting 25 for y in the equation y equals,

open parenthesis, x minus 11, close parenthesis, squared gives 25 equals, open parenthesis, x minus 11, close parenthesis, squared. It follows that

x minus 11, equals 5 or x minus 11, equals negative 5, so the x-coordinates of the two points of intersection are x equals 16 and x equals 6, respectively. Since both points of intersection have a y-coordinate of

25, it follows that the two points are 16 comma 25 and 6 comma 25. Since these points lie on the horizontal line y equals 25, the distance between these points is the positive difference of the x-coordinates: 16 minus 16, equals 10.

Alternatively, since a translation is a rigid motion, the distance between points A and B would be the same as the distance between the points of intersection of the line y equals 25 and the parabola y equals x squared. Since those

graphs intersect at the points with coordinates 0 comma 5 and 0 comma negative 5, the distance between the two points, and thus the distance between A and B, is 10.Incorrect answerChoices B, C, and D are incorrect and may be the result of an error in solving the quadratic equation that results when substituting 25 for y in the given quadratic equation.

Explanation for question 11.Correct answerChoice B is correct. Since the angles marked y degrees and u degrees are vertical angles, y equals u. Substituting y for u in the equation

x plus y, equals u plus w gives x equals w. Since the angles marked w degrees and z degrees are vertical angles, w equals z. Therefore, by the transitive property, x equals z, and so 1 must be true.

The equation in 2 need not be true. For example, if x equals w, which equals z, which equals t, which equals 70 and y equals u, which equals 40, then all three pairs of vertical angles in the figure have equal measure and the given condition x plus y, equals u plus w holds. But it is not true in this case that y is equal to w. Therefore, 2 need not be true.

Since the top three angles in the figure form a straight angle, it follows that x plus y, plus z, equals 180. Similarly, w plus u,

plus t, equals 180, and so x plus y, plus z, equals w plus u, plus t. Subtracting the sides of the given equation x plus y, equals u plus w from the corresponding sides of x plus y, plus z, equals w plus u, plus t gives z equals t. Therefore, 3 must be true. Since only 1 and 3 must be true, the correct answer is choice B.Incorrect answerChoices A, C, and D are incorrect because each of these choices includes 2, which need not be true.


Choice A is correct. The parabola with equation y equals, a, times open parenthesis, x minus 2, close parenthesis, times open parenthesis, x plus

4, close parenthesis crosses the x-axis at the points with coordinates

negative 4 comma 0 and 2 comma 0. By symmetry, the x-coordinate of the

vertex of the parabola is halfway between the x-coordinates of negative 4

comma 0 and 2 comma 0. Thus, the x-coordinate of the vertex is

the fraction with numerator negative 4 plus 2, and denominator 2, equals negative 1. This is the value of c. To find the y-coordinate of the vertex,

substitute negative 1 for x in y equals, a, times open parenthesis, x minus 2, close parenthesis, times open parenthesis, x plus 4, close parenthesis:

y equals, a, times open parenthesis, x minus 2, close parenthesis, times open parenthesis, x plus 4, close parenthesis, which equals a, times open parenthesis, negative 1 minus 2, close parenthesis, times open parenthesis, negative 1 plus 4, which equals a, times negative 3, times 3, which equals negative 9 a.

Therefore, the value of d is negative 9 a.Incorrect answerChoice B is incorrect because the value of the constant term in the equation is not the y-coordinate of the vertex, unless there were no linear terms in the quadratic. Choice C is incorrect and may be the result of a sign error in finding the x-coordinate of the vertex. Choice D is incorrect because the negative of the coefficient of the linear term in the quadratic equation is not the y-coordinate of the vertex.

Explanation for question 13.Correct answerChoice B is correct. Since 24, x squared, plus 25 x, minus 47 divided by a, x minus 2 is equal to negative 8 x minus 3 with

remainder negative 53, it is true that open parenthesis, negative 8 x minus 3, close parenthesis, times open parenthesis, a, x minus 2, close parenthesis, minus 53, equals 24, x squared, plus 25 x, minus 47. (This can be seen by multiplying each side of the given equation by a, x minus 2). This can be rewritten as negative 8 a, x squared, plus 16 x, minus 3 a, x, plus 6, minus 53, which equals 24, x squared, plus 25 x, minus 47. Since the coefficients of the x squared-term have to be equal on both sides of the equation, negative 8 a, equals 24, or a equals negative 3.Incorrect answerChoices A, C, and D are incorrect and may be the result of either a conceptual misunderstanding or a computational error when trying to solve for the value of a.

Explanation for question 14.Correct answerChoice A is correct. Dividing each side of the given equation by 3 gives the equivalent equation x squared, plus 4 x, plus 2, equals 0. Then

using the quadratic formula, the fraction with numerator, negative b, plus or minus the square root of, b squared minus 4 a c, end root, and denominator 2 a, end fraction with a, equals 1, b equals 4, and c equals 2, gives the solutions x equals, negative 2 plus or minus the square root of 2.Incorrect answerChoices B, C, and D are incorrect and may be the result of errors when applying the quadratic formula.


Choice D is correct. If C is graphed against F, the slope of the line is equal to five ninths degrees Celsius/degrees Fahrenheit, which means that for an increase

of 1 degree Fahrenheit, the increase is five ninths of 1 degree Celsius. Thus, statement 1 is true. This is the equivalent to saying that an increase of 1 degree

Celsius is equal to an increase of nine fifths degrees Fahrenheit.

Since nine fifths, equals 1.8, statement 2 is true. On the other hand,

statement 3 is not true, since a temperature increase nine fifths degrees

Fahrenheit, not five ninths degree Fahrenheit, is equal to a temperature increase of 1 degree Celsius.Incorrect answerChoices A, B, and C are incorrect because each of these choices omits a true statement or includes a false statement.

Explanation for question 16.Correct answerThe correct answer is either 1 or 2. The given equation can be rewritten as

x to the fifth power, minus 5 x cubed, plus 4 x, equals 0. Since the polynomial expression on the left has no constant term, it has x as a factor:

x times open parenthesis, x to the fourth power, minus 5 x squared, plus 4, close parenthesis, equals 0. The expression in parentheses is a quadratic equation in x squared that can be factored, giving

x times open parenthesis, x squared minus 1, close parenthesis, times open parenthesis, x squared minus 4, close parenthesis, equals 0.

This further factors as x times open parenthesis, x minus 1, close parenthesis, times open parenthesis, x plus 1, close parenthesis, times open parenthesis, x minus 2, close parenthesis, times open parenthesis, x plus 2, close parenthesis, equals 0. The solutions for x are x equals 0, x equals 1, x equals negative 1, x equals 2, and x equals negative 2. Since it is given that x is greater than 0, the possible values of x are x equals 1 and x equals 2. Either 1 or 2 may be gridded as the correct answer.

Explanation for question 17.Correct answerThe correct answer is 2. First, clear the fractions from the given equation by multiplying each side of the equation by 36 (the least common multiple of 4, 9, and 12). The equation becomes 28 x, minus 16 x, equals 9 plus 15. Combining like terms on each side of the equation yields 12 x equals 24. Finally, dividing both sides of the equation by 12 yields x equals

2. Alternatively, since seven ninths x, minus four ninths x,

equals three ninths x, which equals one third x and one fourth, plus five twelfths, equals three twelfths, plus five twelfths, which equals eight twelfths, which equals two thirds the given equation simplifies to

one third x, equals two thirds. Multiplying each side of one third x, equals two thirds by 3 yields x equals 2.

Explanation for question 18.Correct answerThe correct answer is 105. Since 180 minus z, equals 2 y and y equals 75, it follows that 180 minus x, equals 150, and so z equals 30. Thus, each of the base angles of the isosceles triangle on the right has

measure the fraction with numerator 180 degrees, minus 30 degrees, and denominator 2, equals 75 degrees. Therefore, the measure of the angle marked x degrees is 80 degrees, minus 75 degrees, equals 105 degrees, and so the value of x is 105.

Explanation for question 19.Correct answerThe correct answer is 370. A system of equations can be used where h represents the number of calories in a hamburger and f represents the number of calories in an order of fries. The equation 2 h plus 3 f, equals 1700 represents the fact that 2 hamburgers and 3 orders of fries contain a total of 1700 calories, and the equation h equals f, plus 50 represents the fact that one hamburger contains 50 more calories than an order of fries. Substituting f plus 50 for

h in 2 h plus 3 f, equals 1700 gives 2 times open parenthesis, f plus 50, close parenthesis, plus 3 f, equals 1700. This equation can be solved as follows:

2 f plus 100, plus 3 f, equals 1700

5 f plus 100, equals 1700

5 f equals 1600

f equals 320

The number of calories in an order of fries is 320, so the number of calories in a hamburger is 50 more than 320, or 370.


The correct answer is three fifths or .6. Triangle A B C is a right triangle with its right angle at B. Thus, line segment A C is the hypotenuse of right triangle A B C, and line segments A B and B C are the legs of right triangle A B C. By the Pythagorean theorem,

the length of segment A B equals, the square root of 20 squared, minus 16 squared, end root, which equals the square root of 400 minus 256, end root, which equals the square root of 144, which equals 12. Since triangle D E F is similar to triangle A B C, with vertex F corresponding to vertex C, the measure of angle F equals the measure of angle C. Thus, the sine of F, equals the sine of C. From the side lengths of

triangle A B C, the sine of C, equals opposite side over hypotenuse, which equals the length of segment A B over the length of

segment A C, which equals 12 over 20, which equals three fifths. Therefore, the sine of F equals three fifths. Either 3 slash 5 or its decimal equivalent, .6, may be gridded as the correct answer.

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