Download - SAT Practice Test 3 Math Test: No Calculator, for Assistive Technology

Math Test—No Calculator20 Questions

Turn to Section 3 of your answer sheet to answer the questions in this section.

Directions

For questions 1 through 15, solve each problem, choose the best answer from the choices provided, and fill in the corresponding circle on your answer sheet. For questions 16 through 20, solve the problem and enter your answer in the grid on the answer sheet. Please refer to the directions before question 16 on how to enter your answers in the grid. You may use scratch paper for scratch work.

Notes

1. The use of a calculator is not permitted.

2. All variables and expressions used represent real numbers unless otherwise indicated.

3. Figures provided in this test are drawn to scale unless otherwise indicated.

4. All figures lie in a plane unless otherwise indicated.

5. Unless otherwise indicated, the domain of a given function f is the set of all

real numbers x for which f of x is a real number.

The S A T® Copyright 2015 by the College Board W F-5 L S A 0 8

Reference

Begin skippable figure descriptions.

The figure presents information for your reference in solving some of the problems.


Reference figure 1 is a circle with radius r. Two equations are presented below reference figure 1.

A equals pi times the square of r.

C equals 2 pi r.

Reference figure 2 is a rectangle with length ℓ and width w. An equation is presented below reference figure 2.

A equals ℓ w.

Reference figure 3 is a triangle with base b and height h. An equation is presented below reference figure 3.

A equals one-half b h.

Reference figure 4 is a right triangle. The two sides that form the right angle are labeled a and b, and the side opposite the right angle is labeled c. An equation is presented below reference figure 4.

c squared equals a squared plus b squared.

Special Right Triangles

Reference figure 5 is a right triangle with a 30-degree angle and a 60-degree angle. The side opposite the 30-degree angle is labeled x. The side opposite the 60-degree angle is labeled x times the square root of 3. The side opposite the right angle is labeled 2 x.


Reference figure 6 is a right triangle with two 45-degree angles. Two sides are each labeled s. The side opposite the right angle is labeled s times the square root of 2.


Reference figure 7 is a rectangular solid whose base has length ℓ and width w and whose height is h. An equation is presented below reference figure 7.

V equals ℓ w h.

Reference figure 8 is a right circular cylinder whose base has radius r and whose height is h. An equation is presented below reference figure 8.

V equals pi times the square of r times h.

Reference figure 9 is a sphere with radius r. An equation is presented below reference figure 9.

V equals four-thirds pi times the cube of r.

Reference figure 10 is a cone whose base has radius r and whose height is h. An equation is presented below reference figure 10.

V equals one-third times pi times the square of r times h.

Reference figure 11 is an asymmetrical pyramid whose base has length ℓ and width w and whose height is h. An equation is presented below reference figure 11.

V equals one-third ℓ w h.

End skippable figure descriptions.

Additional Reference Information

The number of degrees of arc in a circle is 360.


The number of radians of arc in a circle is 2 pi.

The sum of the measures in degrees of the angles of a triangle is 180.


Question 1.A painter will paint n walls with the same size and shape in a building using a specific brand of paint. The painter’s fee can be calculated by the expression 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. If the customer asks the painter to use a more expensive brand of paint, which of the factors in the expression would change?

A. h

B. l

C. K

D. n

Explanation for question 1.

Question 2.If 3 r equals 18, what is the value of 6 r plus 3?

A. 6

B. 27

C. 36

D. 39



Question 3.

Which of the following is equal to a, raised to the power two thirds, for all values of a ?

A. the square root of a, raised to the power one third, end root

B. the square root of a, cubed, end root

C. the cube root of a, raised to the power one half, end root

D. the cube root of a, squared, end root


Question 4.The number of states that joined the United States between 1776 and 1849 is twice the number of states that joined between 1850 and 1900. If 30 states joined the United States between 1776 and 1849 and x states joined between 1850 and 1900, which of the following equations is true?

A. 30 x equals 2

B. 2 x equals 30

C. the fraction x over 2 equals 30

D. x plus 30 equals 2



Question 5.

If the fraction 5 over x equals the fraction whose numerator is 15 and

whose denominator is x plus 20 end fraction, what is the value of the fraction x over 5 ?

A.

B.

C.

D. one half


Question 6 refers to the following equations.

2 x minus 3 y equals negative 14

3 x minus 2 y equals negative 6

Question 6.


If the ordered pair x comma y is a solution to the preceding system of equations, what is the value of x minus y ?

A. negative 20

B. negative 8

C. negative 4

D. 8


Question 7 refers to the following table.

x f of x

0 3

2 1

4 0

5−2

negative 2

Question 7.

The function f is defined by a polynomial. Some values of x and f of x are

shown in the preceding table. Which of the following must be a factor of f of x ?


A. x minus 2

B. x minus 3

C. x minus 4

D. x minus 5



Question 8.

The line y equals k x plus 4, where k is a constant, is graphed in the

xy-plane. If the line contains the point with coordinates c comma d, where

c is not equal to 0 and d is not equal to 0, what is the slope of the line in terms of c and d ?

A. the fraction d minus 4, over c

B. the fraction c minus 4, over d

C. the fraction 4 minus d, over c

D. the fraction 4 minus c, over d



Question 9 refers to the following system of equations.

k x minus 3 y equals 4

4 x minus 5 y equals 7

Question 9.In the preceding system of equations, k is a constant and x and y are variables. For what value of k will the system of equations have no solution?

A. the fraction 12 over 5

B. the fraction 16 over 7

C. the negative fraction 16 over 7

D. the negative fraction 12 over 5



Question 10.

In the xy-plane, the parabola with equation y equals, parenthesis,

x minus 11, close parenthesis, squared, intersects the line with equation

y equals 25 at two points, A and B. What is the length of line segment A B?

A. 10

B. 12

C. 14

D. 16



Question 11 refers to the following figure.

Begin skippable figure description.

The figure presents three lines k, l, and m that intersect at a point. The three lines form six angles at the point of intersection. The horizontal line is labeled m, the line that slants upward and to the right is labeled l, and the line that slants downward and to the right is labeled k. Of the 6 angles, 3 angles are above line m, and 3 angles are below line m. Starting from the leftmost angle above line m and going clockwise, the 6 angles are labeled x degrees, y degrees, z degrees, t degrees, u degrees, and w degrees, respectively.

End skippable figure description.


Question 11.

In the preceding figure, lines k, l, and m intersect at a point. If x plus y equals u plus w, which of the following must be true?

I. x equals z

II. y equals w

III. z equals t

A. I and II only

B. I and III only

C. II and III only

D. I, II, and III



Question 12 refers to the following quadratic equation.

y equals a, parenthesis, x minus 2, close parenthesis, times, parenthesis, x plus 4, close parenthesis

Question 12.In the preceding quadratic equation, a is a nonzero constant. The graph of the

equation in the xy-plane is a parabola with vertex with coordinates c comma d. Which of the following is equal to d ?

A. negative 9 a

B. negative 8 a

C. negative 5 a

D. negative 2 a



Question 13.

The equation with the fraction whose numerator is 24 x squared, plus 25 x minus 47, and whose denominator is a, x minus 2, equals, negative 8 x minus 3, minus, the fraction whose numerator is 53, and

whose denominator is a, x minus 2, is true for all values of x that do not equal the fraction, 2 over a, where a is a constant. What is the value of a ?

A. negative 16

B. negative 3

C. 3

D. 16



Question 14.

What are the solutions to 3 x squared, plus 12 x plus 6, equals 0 ?

A. x equals negative 2, plus or minus the square root of 2

B. x equals negative 2, plus or minus, the fraction the square root of 30, end square root, over 3

C. x equals negative 6, plus or minus the square root of 2

D. x equals negative 6, plus or minus, 6 times the square root of 2



Question 15 refers to the following equation.

C equals five ninths, parenthesis F minus 32 close parenthesis

Question 15.The preceding equation shows how a temperature F, measured in degrees Fahrenheit, relates to a temperature C, measured in degrees Celsius. Based on the equation, which of the following must be true?

I. A temperature increase of 1 degree Fahrenheit is equivalent to a temperature

increase of five ninths degree Celsius.

II. A temperature increase of 1 degree Celsius is equivalent to a temperature increase of 1.8 degrees Fahrenheit.

III. A temperature increase of five ninths degree Fahrenheit is equivalent to a temperature increase of 1 degree Celsius.

A. I only

B. II only

C. III only

D. I and II only



Directions

For questions 16 through 20, solve the problem and enter your answer in the grid, as described below, on the answer sheet.

1. Although not required, it is suggested that you write your answer in the boxes at the top of the columns to help you fill in the circles accurately. You will receive credit only if the circles are filled in correctly.

2. Mark no more than one circle in any column.

3. No question has a negative answer.

4. Some problems may have more than one correct answer. In such cases, grid only one answer.

5. Mixed numbers such as three and one half must be gridded as 3.5 or

seven slash two. (If three, one, slash, two, is entered into the grid, it

will be interpreted as thirty one halves, not three and one half.)

6. Decimal answers: If you obtain a decimal answer with more digits than the grid can accommodate, it may be either rounded or truncated, but it must fill the entire grid.


The following are four examples of how to record your answer in the spaces provided. Keep in mind that there are four spaces provided to record each answer.

Examples 1 and 2


Example 1: If your answer is a fraction such as seven-twelfths, it should be recorded as follows. Enter 7 in the first space, the fraction bar (a slash) in the second space, 1 in the third space, and 2 in the fourth space. All four spaces would be used in this example.

Example 2: If your answer is a decimal value such as 2.5, it could be recorded as follows. Enter 2 in the second space, the decimal point in the third space, and 5 in the fourth space. Only three spaces would be used in this example.



Example 3


Example 3: Acceptable ways to record two-thirds are: 2 slash 3, .666, and .667.



Example 4

Note: You may start your answers in any column, space permitting. Columns you don’t need to use should be left blank.


Example 4: It is not necessary to begin recording answers in the first space unless all four spaces are needed. For example, if your answer is 201, you may record 2 in the second space, 0 in the third space, and 1 in the fourth space. Alternatively, you may record 2 in the first space, 0 in the second space, and 1 in the third space. Spaces not needed should be left blank.



Question 16 refers to the following equation.

x cubed, parenthesis, x squared minus 5, close parenthesis, equals negative 4 x

Question 16.If x is greater than 0, what is one possible solution to the preceding equation?


Question 17.

If seven ninths x, minus four ninths x, equals, one fourth plus five twelfths what is the value of x ?



Question 18 refers to the following figure.


The figure presents two triangles that have one vertex in common. The two triangles, one on the left and one on the right, each have a vertical side opposite the common vertex, and the other two sides in each triangle are marked showing that both sides are equal in length. In the triangle on the left, the angle at the common vertex is labeled y degrees, and in the triangle on the right, the angle at the common vertex is labeled z degrees. The exterior angle of the triangle on the right that is between one of the equal sides and the vertical line extended from the vertical side is labeled x degrees.



Question 18.

Two isosceles triangles are shown in the preceding figure. If

180 minus z equals 2 y and y equals 75, what is the value of x ?


Question 19.At a lunch stand, each hamburger has 50 more calories than each order of fries. If 2 hamburgers and 3 orders of fries have a total of 1700 calories, how many calories does a hamburger have?


Question 20.

In triangle A B C, the measure of angle B is 90 degrees, B C equals 16, and A C equals 20. Triangle D E F is similar to triangle A B C, where vertices D, E, and F correspond to vertices A, B, and C, respectively, and each side

of triangle D E F is one third the length of the corresponding side of triangle A B C. What is the value of sine F?



Stop

If you finish before time is called, you may check your work on this section only. Do not turn to any other section.

Answers and explanations for questions 1 through 20 are provided in the next section of this document.


Answers and Explanations for Questions 1 through 20

Explanation for question 1.Choice 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 the walls 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.

Choice 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.Choice 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, parenthesis, 6, close parenthesis, plus 3, equals 39.

Alternatively, the expression 6 r plus 3 can be rewritten as 2(3r) + 3 2, parenthesis, 3 r, close parenthesis, plus 3. Substituting 18 for 3r 3 r in the expression 2(3r) + 3 2, parenthesis, 3 r, close parenthesis, plus 3, yields 2(18) + 3 = 36 + 3 = 39 2, parenthesis 18 close parenthesis, plus 3, equals 36 plus 3, which equals 39.

Choice A is incorrect because 6 is the value of r; however, the question asks for the value of the expression 6r + 36 r plus 3.

Choices B and C are incorrect because if 6r + 3 6 r plus 3 were equal to either of these values, then it would not be possible for 3r 3 r to be equal to 18, as stated in the question.

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 3r to be equal to 18, as stated in the question.



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

m and n. It follows, therefore, that a, raised to the power two thirds, equals, the cube root of a, squared, end root.

Choice A is incorrect. By definition, a, raised to the power the fraction one over n, equals the nth 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, raised to the power one third, end root, yields

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

a, raised to the power one sixth, does not equal a, raised to the power

two thirds, the square root of a, raised to the power one third, end root, is not the correct answer.


Choice B is incorrect. By definition, a, raised to the power the fraction one over n, equals the nth 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 parenthesis, a, cubed, close parenthesis, raised to the power

one half, which equals a, raised to the power three halves. Because a, raised to the power three halves, does not equal a, raised to the power two thirds,

the square root of a, cubed, end root, is not the correct answer.

Choice C is incorrect. By definition, a, raised to the power the fraction one over n, equals the nth 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, raised to the power one half, end root, yields the cube root of a, raised to the power one half, end root, equals, parenthesis, a, raised to the power one half, close parenthesis, raised to the power one third, which

equals a, raised to the power one sixth. Because a, raised to the power

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


Explanation for question 4.Choice 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.

Choices 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 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 whose numerator is 15, and whose denominator is x plus 20, end fraction by x, parenthesis, x plus 20, close parenthesis, gives

15 x equals, 5, parenthesis, x plus 20, close parenthesis. Distributing the 5 over the values within the parentheses yields 15 x equals 5 x plus 100, and then subtracting 5 x from each side gives 10 x equals 100. Finally, dividing both sides by 10 gives x equals 10.

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

Choice A is incorrect because it is the value of x, not the fraction 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.Choice 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 −5 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, parenthesis, 6, close parenthesis, 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.

Choices A, B, and D are incorrect and may be the result of an arithmetic error when solving the system of equations.


Explanation for question 7.Choice C is correct. If x minus b is a factor of f(x) f of x, then f(b) 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(x) f of x.

Choice A is incorrect because f of 2 does not equal 0, choice B is incorrect because no information is given about the value of f(3) f of 3, so x minus 3 may or may not be a factor of f(x) f of x, and choice D is incorrect because f of 5 does not equal 0.

Explanation for question 8.Choice 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 (c, d) with coordinates c comma d, the coordinates of this point satisfy the equation y equals k x plus 4: d equals k c plus 4. Solving this

equation for the slope, k, gives k equals the fraction d minus 4, over c.

Choices B, C, and D are incorrect and may be the result of errors in substituting the coordinates of (c, d) 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.Choice 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

kx minus 3 y equals 4 can be rewritten as y equals the

fraction k over 3, end fraction, times x, minus, the fraction 4 over 3, 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 the fraction 4 over 5, times x,

minus the fraction 7 over 5, where the fraction 4 over 5 is the slope of the line. If

two lines are parallel, then the slopes of the line are equal. Therefore, the

fraction 4 over 5 equals, the fraction k over 3, or k equals the fraction 12

over 5. (Since the y-intercepts of the lines represented by the equations are the

negative fraction 4 over 3 and the negative fraction 7 over 5, the lines are parallel, not identical.)

Choices 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

parenthesis, x minus 11, close parenthesis, squared gives 25 equals 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, 25) 16 comma 25 and (6, 25) 6 comma 25. Since these points lie on the horizontal line y = 25 y equals 25, the distance between these points is the positive difference of the x-coordinates:

16 minus 6 equals 10.

Choices 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.Choice B is correct. Since the angles marked y degrees and u degrees are vertical angles, y equals u. Subtracting the sides of from the corresponding sides of 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, x equals z, and so statement I must be true.

The equation in statement II 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, statement II 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, statement III must be true. Since statements I and III only must be true, the correct answer is B.

Choices A, C, and D are incorrect because each of these choices includes statement II, which need not be true.

Explanation for question 12.Choice A is correct. The parabola with equation y equals a, parenthesis, x minus 2, close parenthesis, times, parenthesis, x plus 4, close parenthesis, crosses the x-axis at the points (−4, 0) negative 4 comma 0 and (2, 0) 2 comma 0. The x-coordinate of the vertex of the parabola is halfway between the x-coordinates of (−4, 0) negative 4 comma 0 and (2, 0) 2 comma 0. Thus, the

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

y equals a, parenthesis, x minus 2, close parenthesis, times, parenthesis, x plus 4, close parenthesis:


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

Therefore, the value of d is −9a negative 9 a.

Choice 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 term 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 is not the y-coordinate of the vertex.

Explanation for question 13.Choice 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 parenthesis, negative 8 x minus 3, close parenthesis, times, 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, 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.

Choices 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.Choice 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

quadratic formula, the fraction whose numerator is negative b plus or minus, the square root of b squared minus 4 a c, end square root, and whose denominator is 2 a 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.

Choices 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 graph 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 I 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 II is true. On the other hand, statement III is not

true, since a temperature increase of nine fifths degrees Fahrenheit, not five ninths degree Fahrenheit, is equal to a temperature increase of 1 degree Celsius.


Choice A, B, and C are incorrect because each of these choices omits a true statement or includes a false statement.


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

x raised to the power 5, end 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, parenthesis, x raised to the power 4, end 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, parenthesis, x squared minus 1, close parenthesis, times, parenthesis, x squared minus 4, close parenthesis, equals 0. This further factors as

x, parenthesis, x minus 1, close parenthesis, times, parenthesis, x plus 1, close parenthesis, times, parenthesis, x minus 2, close parenthesis, times, 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.The 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, 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.The correct answer is 105. Since 180 minus z equals 2 y, and y equals 75, it follows that 180 minus z equals 150, and so z equals 30. Thus, each of the base angles of the isosceles triangle on the right has

measure the fraction 180 degrees minus 30 degrees, over 2, equals 75 degrees. Therefore, the measure of the angle marked x degrees is

180 degrees minus 75 degrees, equals 105 degrees, and so the value of x is 105.


Explanation for question 19.The 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 1,700 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 1,700 gives 2, parenthesis, f plus 50, close parenthesis, plus 3 f, equals 1,700. This equation can be solved as follows:

2 f plus 100 plus 3 f equals 1,700

5 f plus 100 equals 1,700

5 f equals 1,600

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 segment A B and line segment B C are the legs of right triangle A B C. By the Pythagorean theorem,

A B equals, the square root of 20 squared minus 16 squared, end square root, which equals, the square root of 400 minus 256, end square root, which equals the square root of 144, which equals 12. Since triangle DEF 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,

sine F equals sine C. From the side lengths of triangle A B C,

sine C equals the fraction opposite side over hypotenuse, which equals the fraction A B over A C, which equals the fraction

12 over 20, which equals three fifths. Therefore, sine F equals three

fifths. Either three fifths or its decimal equivalent, .6, may be gridded as the correct answer

Stop. This is the end of the answers and explanations for questions 1 through 20.


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