If the expression 
, then one possible value of x could be:
- A . C 1
- B . C 2
- C . C 5
- D . 1
- E . 2
A
Explanation:
Cross multiply and solve for x:
3 × 2x = ( 2 + x) × (x C 5)
6x = x 2 C 3x C 10
x 2 C 9x C 10 = 0
(x C 10) × (x + 1)
x = 10, x = C1
In the graph above, ABCD is a square.
What are the coordinates of point B?
- A . (C1, C4)
- B . (C1, 4)
- C . (C1, 6)
- D . (C3, 1)
- E . (C3, 4)
B
Explanation:
Point B is the same distance from the y-axis as point A, so the x-coordinate of point B is the same as the x-coordinate of point A (C1). Point B is the same distance from the x-axis as point C, so the y-coordinate of point B is the same as the y-coordinate of point C (4). The coordinates of point B are (C1, 4).
Line y = 2/3x C 5 is perpendicular to line:
- A . y = 2/3x + 5
- B . y = 5 C 2/3x
- C . y = C2/3x C 5
- D . y = 2/3x C 5
- E . y = C2/3x + 5
E
Explanation:
Perpendicular lines have slopes that are negative reciprocals of each other. The slope of the line given is 2/3. The negative reciprocal of 2/3 is 3/2. Every line with a slope of C3/ 2 is perpendicular to the given line; y = C3/2x + 5 is perpendicular to y = 2/3x C 5.
If 30% of r is equal to 75% of s, what is 50% of s if r = 30?
- A . 4.5
- B . 6
- C . 9
- D . 12
- E . 15
B
Explanation:
If r = 30, 30% of r = 0.30 × 30 = 9. 9 is equal to 75% of s. If 0.75s = 9, then s = 12. 50% of s = 0.50 × 12 =
If 30% of r is equal to 75% of s, what is 50% of s if r = 30?
- A . 4.5
- B . 6
- C . 9
- D . 12
- E . 15
B
Explanation:
If r = 30, 30% of r = 0.30 × 30 = 9. 9 is equal to 75% of s. If 0.75s = 9, then s = 12. 50% of s = 0.50 × 12 =
A dormitory now houses 30 men and allows 42 square feet of space per man.
If five more men are put into this dormitory, how much less space will each man have?
- A . 5 square feet
- B . 6 square feet
- C . 7 square feet
- D . 8 square feet
- E . 9 square feet
B
Explanation:
30 men × 42 square feet = 1260 square feet of space; 1260 square feet ÷ 35 men = 36 square feet; 42 C 36 = 6, so each man will have 6 less square feet of space.
Rob has six songs on his portable music player.
How many different four-song orderings can Rob create?
- A . 30
- B . 60
- C . 120
- D . 360
- E . 720
D
Explanation:
The order of the four songs is important. The orderings A, B, C, D and A, C, B, D contain the same four songs, but in different orders. Both orderings must be counted. The number of six-choose-four orderings is equal to 6 × 5 × 4 × 3 = 360.
The statement “Raphael runs every Sunday” is always true.
Which of the following statements is also true?
- A . If Raphael does not run, then it is not Sunday.
- B . If Raphael runs, then it is Sunday.
- C . If it is not Sunday, then Raphael does not run.
- D . If it is Sunday, then Raphael does not run.
- E . If it is Sunday, it is impossible to determine if Raphael runs.
A
Explanation:
The statement “Raphael runs every Sunday” is equivalent to “If it is Sunday, Raphael runs.” The contra positive of a true statement is also true. The contra positive of “If it is Sunday, Raphael runs” is “If Raphael does not run, it is not Sunday.”
In the diagram above, lines EF and GH are parallel, and line AB is perpendicular to lines EF and GH.
What is the length of line AB?
- A . 5
- B .
- C .
- D .
- E .
C
Explanation:
Line AB is perpendicular to line BC, which makes triangle ABC a right triangle.
and
are
alternating angles, i.e. angles made by a pair of parallel lines cut by a
transversal.
, therefore,
.
form a line. There
are 180º in a line, so the measure of of
. Triangle ABC is a 30-60-90 right triangle, which means that the length of the hypotenuse, AC, is equal to twice the length of the leg opposite the 30-degree angle, BC. Therefore, the length of BC is 10/2, or 5. The length of the leg opposite the 60-degree angle, AB, is √3 times the length of the other leg, BC. Therefore, the length of AB is 5√3.
The expression 
is equivalent to:
- A . 5/7
- B . x + 5
- C . (x + 5) / (x + 7)
- D . C5 / (2x C 7)
- E . (2x C 15) / (4x C 21)
C
Explanation:
Factor the numerator and denominator and cancel like factors:
(x 2 + 2x C 15) = (x + 5) × (x C 3)
(x 2 + 4x C 21) = (x + 7) × (x C 3)
Cancel the (x C 3) term from the numerator and the denominator. The fraction reduces to (x + 5) / (x + 7).
The point (2, 1) is the midpoint of a line with endpoints at (C5, 3) and:
- A . (C3, 4)
- B . (C7, 2)
- C . (7, 1)
- D . (9, C1)
- E . (C10, 3)
D
Explanation:
The midpoint of a line is equal to the average x-coordinates and the average y-coordinates of the line’s endpoints:
(C5 + x) / 2 = 2, C5 + x = 4, x = 9
(3 + y) / 2 = 1, 3 + y = 2, y = C1
The other endpoint of this line is at (9, C1).
Lindsay grows only roses and tulips in her garden. The ratio of roses to tulips in her garden is 5:6.
If there are 24 2 total flowers in her garden, how many of them are tulips?
- A . 22
- B . 40
- C . 110
- D . 121
- E . 132
E
Explanation:
The number of roses, 5x, plus the number of tulips, 6x, is equal to 242 total flowers: 5x + 6x = 242, 11x = 242, x = 22. There are 5 × 22 = 110 roses and 6 × 22 = 132 tulips in Lindsay’s garden.
It takes eight people 12 hours to clean an office.
How long would it take six people to clean the office?
- A . 9 hours
- B . 15 hours
- C . 16 hours
- D . 18 hours
- E . 24 hours
C
Explanation:
There is an inverse relationship between the number of people and the time needed to clean the office. Multiply the number of people by the hours needed to clean the office: 8 × 1 2 = 96. Divide the total number of hours by the new number of people, 6: 96 ÷ 6 = 16. It takes six people 16 hours to clean the office.
Greg has nine paintings. The Hickory Museum has enough space to display three of them.
From how many different sets of three paintings does Greg have to choose?
- A . 27
- B . 56
- C . 84
- D . 168
- E . 504
C
Explanation:
Be careful not to count the same set of three paintings more than once C order is not important. A ninechoose-three combination is equal to
If the surface area of a cube is 384 cm2, what is the volume of the cube?
- A . 64 cm3
- B . 256 cm3
- C . 51 2 cm3
- D . 115 2 cm3
- E . 4096 cm3
C
Explanation:
The surface area of a cube is equal to 6 × e 2 , where e is the length of one edge of the cube; 6 × e 2 = 384 cm, e 3 = 64, e = 8 cm. The volume of a cube is equal to e 3 ; (8 cm) 3 = 512 cm3.
In the diagram above, what is the sum of the measures of the 
, 
and
?
- A . 180º
- B . 360
- C . 540º
- D . 720º
- E . Cannot be determined.
B
Explanation:
There are 180º in a line: ( x + (supplement of
)) + ( y + (supplement of
)) + ( z + (supplement of
)) = 540. The supplement of
, the supplement of
, and the supplement of
are the interior angles of a
triangle. There are 180º in a triangle, so those supplements sum to 180. Therefore,x + y + z + 180 = 540, and x + y + z = 360.
Given the following figure with one tangent and one secant drawn to the circle, what is the measure of 
?
- A . 50
- B . 85
- C . 60
- D . 110
- E . 25
E
Explanation:
The measure of an angle in the exterior of a circle formed by a tangent and a secant is equal to half the difference of the intercepted arcs. The two intercepted arcs are
which is 60°, and
which is
110°. Find half of the difference of the two arcs: 1/2 × (110 C 60) = 1/ 2 × 50 = 25º.
Balloons are sold according to the chart above. If a customer buys one balloon at a time, the cost is $1.00 per balloon. If a customer buys ten balloons at a time, the cost is $0.90 per balloon.
If Carlos wants to buy 2,000 balloons, how much money does he save by buying 1,000 balloons at a time rather than ten balloons at a time?
- A . $200
- B . $300
- C . $500
- D . $600
- E . $800
D
Explanation:
If Carlos buys ten balloons, he will pay 10 × $0.90 = $9. In order to total 2,000 balloons, Carlos will have to make this purchase 2,000 ÷ 10 = 200 times. It will cost him a total of 200 × $9 = $1,800. If Carlos buys 1,000 balloons, he will pay 1,000 × $0.60) = $600. In order to total 2,000 balloons, Carlos will have to make this purchase 2,000 ÷ 1,000 = 2 times. It will cost him a total of 2 × $600 = $1,200. It will save Carlos $1,800 C $1,200 = $600 to buy the balloons 1,000 at a time.
If 
anda and c are doubled, what happens to the value of d ?
- A . The value of d remains the same.
- B . The value of d is doubled.
- C . The value of d is four times greater.
- D . The value of d is halved.
- E . The value of d is four times smaller.
A
Explanation:
If a and c are doubled, the fraction on the left side of the equation becomes
. The fraction has been multiplied by 2/2 which is equal to 1.
Multiplying a fraction by 1 does not change its value:
.
The value of d remains the same.
In the diagram above, line OA is congruent to line OB.
What is the measure of arc CD?
- A . 27.5º
- B . 55º
- C . 70º
- D . 110º
- E . 125º
C
Explanation:
Triangle AOB is isosceles because line OA is congruent to line OB.
and
are both
55º, which means that
= 180 C (55 + 55) = 70º.
is a
central angle and arc CD is its intercepted arc. A central angle and its
intercepted arc are equal in measure, so the measure of arc CD is 70º.
The expression 
is equivalent to:
- A .
- B .
- C .
- D .
- E .
E
Explanation:
Simplify the numerator:
. Simplify the denominator:
. Divide the numerator and denominator by 2:
.
What is the next number in the series below?
3 16 6 12 12 8
- A . 4
- B . 15
- C . 20
- D . 24
- E . 32
D
Explanation:
This series actually has two alternating sets of numbers. The first number is doubled, giving the third number. The second number has 4 subtracted from it, giving it the fourth number.
Therefore, the blank space will be 12 doubled, or 24.
The volume of a glass of water placed in the sun decreases by 20%.
If there are 240 ml of water in the glass now, what was the original volume of water in the glass?
- A . 19 2 ml
- B . 260 ml
- C . 288 ml
- D . 300 ml
- E . 360 ml
D
Explanation:
The original volume of water, x, minus 20% of x, 0.20x, is equal to the current volume of water, 240 ml.
x C 0.20x = 240 ml
The volume of a glass of water placed in the sun decreases by 20%.
If there are 240 ml of water in the glass now, what was the original volume of water in the glass?
- A . 19 2 ml
- B . 260 ml
- C . 288 ml
- D . 300 ml
- E . 360 ml
D
Explanation:
The original volume of water, x, minus 20% of x, 0.20x, is equal to the current volume of water, 240 ml.
x C 0.20x = 240 ml
What is the tenth term of the pattern below?
2/3, 4/9, 8/27, 16/81 …
- A . 20/30
- B . 210/3
- C . 2/310
- D . (2/3)2/3
- E . (2/3)10
E
Explanation:
Each term in the pattern is equal to the fraction 2/3 raised to an exponent that is equal to the position of the term in the sequence. The first term in the sequence is equal to (2/3) 1 , the second term is equal to (2/3) 2 , and so on. Therefore, the tenth term in the sequence will be equal to (2/3) 10 .
How does the area of a rectangle change if both the base and the height of the original rectangle are tripled?
- A . The area is tripled.
- B . The area is six times larger.
- C . The area is nine times larger.
- D . The area remains the same.
- E . The area cannot be determined.
C
Explanation:
Since both dimensions are tripled, there are two additional factors of 3. Therefore, the new area is 3 × 3 = 9 times as large as the original. For example, use a rectangle with a base of 5 and height of 6. The area is 5 × 6 = 30 square units. If you multiply each side length by 3, the new dimensions are 15 and 18. The new area is 15 × 18, which is 270 square units. By comparing the new area with the original area, 270 square units is nine times larger than 30 square units; 30 × 9 = 270.
The equation 
is undefined when x =:
- A . C9
- B . C2
- C . C6
- D . 0
- E . 9
A
Explanation:
An equation is undefined when the value of a denominator in the equation is equal to zero.
Set x 2 + 7 x C 18 equal to zero and factor the quadratic to find its roots:
x 2 + 7 x C 18
( x + 9) × ( x C 2) = 0
x = C9, x = 2
In the diagram above, 
is
congruent to
, and 
is
congruent to
.
If the ratio of the length of AB to the length of EB is 5:1, and the area of triangle BED = 5 a 2 + 10, what is area of triangle ABC?
- A . 5a 2 + 10
- B . 25a 2 + 50
- C . 25a 2 + 100
- D . 125a 2 + 250
- E . Cannot be determined.
D
Explanation:
Triangles ABC and BED have two pairs of congruent angles. Therefore, the third pair of angles must be congruent, which makes these triangles similar. If the area of the smaller triangle, BED, is equal to bh/2, then the area of the larger triangle, ABC, is equal to
or
. The area of triangle ABC is 25 times larger than the area of triangle BED. Multiply the area of triangle BED by 25: 25 × (5 a 2 + 10) = 125 a 2 + 250.
The number p is greater than 0, a multiple of 6, and a factor of 180.
How many possibilities are there for the value of p?
- A . 7
- B . 8
- C . 9
- D . 10
- E . 11
B
Explanation:
The positive factors of 180 (the positive numbers that divide evenly into 180) are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180. Of these numbers, 8 (6, 12, 18, 30, 36, 60, 90, and 180) are multiples of 6.
If g > 0 and h < 0, which of the following is always positive?
- A . g × h
- B . g + h
- C . g C h
- D . |h| C |g|
- E . hg
C
Explanation:
A positive number minus a negative number will not only always be a positive number, but will also be a positive number greater than the first operand. g × h will always be negative when one multiplicand is positive and the other is negative. g + h will be positive when the absolute value of g is greater than the absolute value of h, but g + h will be negative when the absolute value of g is less than the absolute value of h. |h| C |g| will be positive when |h| is greater than g, but |h| C |g| will be negative when |h| is less than g. hg will be positive when g is an even, whole number, but negative when g is an odd, whole number.
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The length of a room is three more than twice the width of the room. The perimeter of the room is 66 feet.
What is the length of the room?
Explanation:
If x is the width of the room, then 3 + 2x is the length of the room. The perimeter is equal to x + x + (3 + 2x) + (3 + 2x) = 66; 6x + 6 = 66; 6x = 60; x = 10. The length of the room is equal to 2x + 3, 2 × 10 + 3 = 23 feet.
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In the diagram above, lines K and L are parallel, and lines M and N are parallel. If b = 8, then a = ____.
The labeled angle formed by lines M and K and the supplement of the labeled angle formed by lines L and N are alternating angles. Therefore, they are congruent.
The angle labeled (10a + 5) and its supplement, which is equal to (8b + 1), total 180º: (10a + 5) + (8b + 1) = 180. If b = 8, then:
(10a + 5) + (8 × 8 + 1) = 180
10a + 70 = 180
10a = 110
a = 11
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If 6x + 9y C 15 = C6, what is the value of C2x C 3y + 5?
Explanation:
The first expression, 6x + 9y C 15, is C3 times the second expression, C2x C 3y + 5 (multiply each term in the second expression by C3 and you’d get the first expression). Therefore, the value of the first expression, C6, is C3 times the value of the second expression. So, you can find the value of the second expression by dividing the value of the first expression byC3: (C6) ÷ (C3) = 2. The value of C3x C 3y + 5 × 2 is just C1/3 times the value of 6x + 9y C 15 × (C6) since C2x C 3y + 5 itself is C1/3 times 6x + 9y C 15.
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Find the measure of 
Explanation:
Triangle DBC and triangle DEF are isosceles right triangles, which means the measures of
and
both equal 45°;
;
;
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If the distance from point (C2, m) to point (4, C1) is 10 units, what is the positive value of m?
Explanation:
First, use the distance formula to form an equation that can be solved for m :
m 2 + 2 m C 63 = 0
Now factor m 2 + 2 m C 63:
( m + 9) × ( m C 7) = 0
m = 7, m = C9.
The positive value of m is 7.
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If 
, then a = 3 when z = ___.
Explanation:
Substitute 3 for a :
. To solve for z , raise both sides of the equation to the power 2/3:
,
.
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The length of a rectangular prism is four times the height of the prism and one-third the width of the prism. If the volume of the prism is 384 in3, what is the width of the prism?
Explanation:
If the height of the prism is h, then the length of the prism is four times that, 4h. The length is one-third of the width, so the width is three times the length: 12h.
The volume of the prism is equal to its length multiplied by its width multiplied by its height:
h × 4h × 12h
48h 3 = 384
h 3 = 8
h = 2
The height of the prism is 2 in, the length of the prism is 2 in × 4 = 8 in, and the width of the prism is 8 in × 3 = 24 in.
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If 2a 2 + b = 10 and C (b/4 + 3a) = 11, what is the positive value of a?
Explanation:
Solve 2 a 2 + b = 10 for b : b = 10 C 2 a 2 .
Substitute (10 C 2 a 2 ) for b in the second equation and solve for a :
C10 + 2 a 2 + 12 a = 44
2 a 2 + 12 a C 54 = 0
2 a C 6 = 0, a = 3
a + 9 = 0, a = C9
The positive value of a is 3.
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Stephanie buys almonds at the grocery store for $1.00 per pound. If she buys 4 pounds of almonds and pays a 5% tax on her purchase, what is Stephanie’s total bill?
Explanation:
If one pound of almonds costs $1.00, then 4 pounds of almonds costs 4 × $1.00 = $4.00. If Stephanie pays a 5% tax, then she pays $4.00 × 0.05 = $0.20 in tax. Her total bill is $4.00 + $0.20 = $4.20.
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The ratio of the number of linear units in the circumference of a circle to the number of square units in the area of that circle is 2:5.
What is the radius of the circle?
Explanation:
The circumference of a circle = 2πr and the area of a circle = πr 2 . If the ratio of the number of linear units in the circumference to the number of square units in the area is 2:5, then five times the circumference is equal to twice the area:
5 × 2πr = 2(πr) 2
10πr = 2πr 2
10r = 2r 2
5r = r 2
r = 5
The radius of the circle is equal to 5.
Which of the following number pairs is in the ratio 4:5?
- A . 1/4, 1/5
- B . 1/5, 1/4
- C . 1/5, 4/5
- D . 4/5, 5/4
- E . 1, 4/5
B
Explanation:
Two numbers are in the ratio 4:5 if the second number is 5/4 times the value of the first number; 1/4 is 5/4 times the value of 1/5.
When x= C3, the expression C2x 2 + 3x C 7 =:
- A . C34
- B . C27
- C . C16
- D . C10
- E . 2
A
Explanation:
Substitute C3 for x:
C2 × (C3) 2 + 3 × (C3) C 7 = C 2 × 9 C 9 C 7 = C18 C 16 = C34
What is the slope of the line C3y = 12x C 3?
- A . C4
- B . C3
- C . 1
- D . 4
- E . 12
A
Explanation:
First, convert the equation to slope-intercept form:
y = mx + b. Divide both sides of the equation by C3:
y = C4x + 1
The slope of a line written in this form is equal to the coefficient of the x term. The coefficient of the x term is C4, so the slope of the line is C4.
In the diagram above, ABCD is a square with an area of 100 cm 2 and lines BD and AC are the diagonals of ABCD.
If line EF is parallel to line BC and the length of line CF = 3√ 2 cm, which of the following is equal to the shaded area?
- A . 25 cm2
- B . 39 cm2
- C . 64 cm2
- D . 78 cm2
- E . 89 cm2
B
Explanation:
The area of a square is equal to S 2 where S is the length of one side of the square. A square with an area of 100 cm 2 has sides that are each equal to √100 = 10 cm. The diagonal of a square is equal to √2 times the length of a side of the square. Therefore, the lengths of diagonals AC and BD are 10√2 cm. Diagonals of a square bisect each other at right angles, so the lengths of segments OB and OC are each 5√ 2 cm. Since lines BC and EF are parallel and lines OC and OB are congruent, lines BE and CF are also congruent. The length of line OF is equal to the length of line OC plus the length of line CF: 5√2 + 3√ 2 = 8√ 2 cm. In the same way, OE = OB + BE = 5√2 + 3√2 = 8 √2 cm. The area of a triangle is equal to 1/2bh, where b is the base of the triangle and h is the height of the triangle. EOF is a right triangle, and its area is equal to 1/2 × 8√2 × 2√8 = 1/2 × 64 × 2 = 64 cm2. The size of the shaded area is equal to the area of EOF minus one-fourth of the area of ABCD: 64 C 1/4 × 100 = 64 C 25 = 39 cm 2 .
If 0.34 < x < 0.40 and 5/16 < x < 9/20, which of the following could be x?
- A . 1/3
- B . 2/5
- C . 3/8
- D . 3/7
- E . 4/9
C
Explanation:
5/16 = 0.3125 and 9/20 = 0.45; 3/8 = 0.375 which is between 0.34 and 0.40, and between 0.3125 and 0.45.
A store prices a coat at $85. During a sale, the coat is sold at 20% off. After the sale, the store raises the price of the coat 10% over its sale price.
What is the price of the coat now?
- A . $18.70
- B . $61.20
- C . $68.00
- D . $74.80
- E . $93.50
D
Explanation:
20% of $85 = 0.20 × $85 = $17.While on sale, the coat is sold for $85 C $17 = $68; 10% of $68 = 0.10 × $68 = $6.80. After the sale, the coat is sold for $68 + $6.80 = $74.80.
The expression 4x 2 C 2x + 3 is equal to 3 when x = 0 and when x =
- A . C1/2
- B . C1/4
- C . 1/8
- D . 1/4
- E . 1/2
E
Explanation:
Set the expression 4x 2 C 2x + 3 equal to 3 and solve for x:
4x 2 C 2x + 3 = 3
4x 2 C 2x + 3 C 3 = 3 C 3
4x 2 C 2x = 0
4x × (x C 1/2) = 0
x = 0, x = 1/2
A spinner is divided into eight equal regions, labeled one through eight.
If Jenna spins the wheel, what is the probability that she will spin a number that is less than four and greater than two?
- A . 1/8
- B . 9/32
- C . 3/8
- D . 1/2
- E . 3/4
A
Explanation:
There are three numbers on the wheel that are less than four (1, 2, 3), but only one of those numbers (3) is greater than two. The probability of Jenna spinning a number that is both less than 4 and greater than 2 is 1/8.
The length of an edge of a cube is equal to half the height of a cylinder that has a volume of 160π cubic units.
If the radius of the cylinder is 4 units, what is the surface area of the cube?
- A . 64 square units
- B . 96 square units
- C . 100 square units
- D . 125 square units
- E . 150 square units
E
Explanation:
The volume of a cylinder is equal to πr2h. The volume of the cylinder is 160π and its radius is 4. Therefore, the height of the cylinder is equal to:
160π = π × 4 2 × h
160 = 16h
h = 10
The length of an edge of the cube is equal to half the height of the cylinder. The edge of the cube is 5 units. The surface area of a cube is equal to 6e2, where e is the length of an edge of the cube. The surface area of the cube = 6 × 5 2 = 6 × 25 = 150 square units.
The function m#n is equal to m 2 C n.
Which of the following is equivalent to m#(n#m)?
- A . Cn
- B . n 2 C m
- C . m2+ m C n 2
- D . (m 2 C n) 2 C n
- E . (n 2 C m) 2 C m
C
Explanation:
m#n is a function definition. The problem is saying “m#n” is the same as “m 2 C n”. If m#n is n 2 C n, then n#m is n2C m. So, to find m#(n#m), replace (n#m) with the value of (n#m), which is n 2 C m: m#(n 2 C m).
Now, use the function definition again. The function definition says “take the value before the # symbol, square it, and subtract the value after the # symbol”: m squared is m 2 minus the second term, (n 2 C m), is equal to m 2 C (n 2 C m) = m 2 C n 2 + m.
Which of the following has the greatest value when x = C1/4?
- A . x C1
- B . C3/8x
- C . 4x + 3
- D . 16x
- E . 1/81x
E
Explanation:
x C1 = 1/x = 1 ÷ (C1/4) = C4;
C3/8x = C3 ÷ 8 × (C1/4) = 3/2;
4x + 3 = 4 × (C1/4) + 3 = C1 + 3 = 2;
16x = 16 (C1/4) = 1 ÷ 161/4 = 1/2;
1/81x = 1 ÷ 81 (C1/4) = 811/4 = 3.
In the diagram above, lines M and N are parallel.
All of the following are true EXCEPT:
- A . a + b = j + l.
- B . g = h.
- C . c + f = f + b.
- D . g + e + f + h = 360.
- E . d + e = f + j.
E
Explanation:
and 
are vertical angles, so 
. However, 
and
are not alternating angles.
These angles are formed by different transversals. It cannot be stated that
, therefore, it cannot be stated that d + e = f + j.
Melissa runs the 50-yard dash five times, with times of 5.4 seconds, 5.6 seconds, 5.4 seconds, 6.3 seconds, and 5.3 seconds.
If she runs a sixth dash, which of the following would change the mean and mode of her scores, but not the median?
- A . 5.3 seconds
- B . 5.4 seconds
- C . 5.5 seconds
- D . 5.6 seconds
- E . 6.3 seconds
A
Explanation:
Melissa’s mean time for the first five dashes is
. Her times, in order from least to greatest, are: 5.3, 5.4, 5.4, 5.6, and 6.3. The middle score, or median, is 5.4. The number that appears most often, the mode, is 5.4. A score of 5.3 means that the mean will decrease and that the mode will no longer be 5.4 alone. The mode will now be 5.3 and 5.4. The median, however, will remain 5.4.
If 
and
, 
___.
- A . x/y + 1.
- B . x/y + x.
- C . x/y + y.
- D . 2xy
- E . y2 + x.
B
Explanation:
The scatterplot above shows the speeds of different runners over time.
Which of the following could be the equation of the line of best fit?
- A . S = C2 × (t C 15)
- B . S = Ct + 25
- C . S = C1/2 × (t C 10)
- D . S = C1/2 × (t C 20)
- E . S = 2 × (t + 15)
A
Explanation:
If a straight line were drawn through as many of the plotted points as possible, it would have a negative slope. The line slopes more sharply than the line y = Cx × (a line with a slope of C1), so the line would have a slope more negative than C1. The line would also have a y-intercept well above the x-axis. The only equation given with a slope more negative than C1 is S = C2 × (t C 15).
The radius of the outer circle shown above is 1.2 times greater than the radius of the inner circle.
What is the area of the shaded region?
- A . 6π m2
- B . 9π m2
- C . 25π m2
- D . 30π m2
- E . 36π m2
B
Explanation:
The area of a circle is equal to πr2. The radius of the inner circle is 5 m; therefore, the area of the inner circle is 25π m2. The radius of the outer circle is 1. 2 × 5 = 6 m; therefore, the area of the outer circle is 36π. Subtract the area of the inner circle from the area of the outer circle: 36π C 25π = 9π m2.
If m = 6, then the expression 
is equal
to:
- A . C12.
- B . C2.
- C . 6.
- D . 12.
- E . 22.
B
Explanation:
Substitute 6 for m:
Which of the following is the midpoint of a line with endpoints at (C2, C8) and (8, 0)?
- A . (3, 4)
- B . (3, C4)
- C . (C5, 4)
- D . (5, C4)
- E . (6, C8)
B
Explanation:
The midpoint of a line is equal to the average of the x – and y -coordinates of its endpoints. The average of the x -coordinates
. The average of the y -coordinates
. The midpoint of this line is at (3, -4)
If 4x + 5 = 15, then 10x + 5 =:
- A . 2.5.
- B . 15.
- C . 22.5.
- D . 25.
- E . 30.
E
Explanation:
If 4x + 5 = 15, then 4x = 10 and x = 2.5. Substitute 2.5 for x in the second equation: 10 × 2.5 + 5 = 25 + 5 = 30.
A music store offers customized guitars. A buyer has four choices for the neck of the guitar, two choices for the body of the guitar, and six choices for the color of the guitar.
The music store offers:
- A . 1 2 different guitars.
- B . 16 different guitars.
- C . 24 different guitars.
- D . 36 different guitars.
- E . 48 different guitars.
E
Explanation:
To find the total number of different guitars that are offered, multiply the number of neck choices by the number of body choices by the number of color choices: 4 × 2 ×6 = 48 different guitars.
Which of the following is the set of positive factors of 1 2 that are NOT multiples of 2?
- A . { }
- B . {1}
- C . {1, 3}
- D . {1, 2, 3}
- E . {2, 4, 6, 12}
C
Explanation:
The set of positive factors of 1 2 is {1, 2, 3, 4, 6, 12}. All of the even numbers (2, 4, 6, and 12) are multiples of 2. The only positive factors of 1 2 that are not multiples of 2 are 1 and 3.
The graph of f(x) is shown above.
How many values can be found for f(3)?
- A . 0
- B . 1
- C . 2
- D . 4
- E . Cannot be determined.
B
Explanation:
Be careful C the question asks you for the number of values of f(3) not f(x) = 3. In other words, how many y values can be generated when x = 3? If the line x = 3 is drawn on the graph, it passes through only one point. There is only one value for f(3).
The expression 
can be reduced to:
- A . 1.
- B .
- C . x + 5.
- D .
- E .
D
Explanation:
Factor the numerator and denominator of the fraction:
(x 2 + 5x) = x × (x + 5)
(x3C 25x) = x × (x + 5) × (x C 5)
There is an x term and an (x + 5) term in both the numerator and denominator. Cancel those terms, leaving the fraction
.
Which of the following is the vertex of the parabola which is the graph of the equation y = (x + 1)2 + 2?
- A . (C1, C2)
- B . (1, C2)
- C . (C1, 2)
- D . (1, 2)
- E . (2, C1)
C
Explanation:
The equation of a parabola with its turning point c units to the left of the y-axis is written as y = (x + c)2. The equation of a parabola with its turning point d units above the x-axis is written as y = x2 + d. The vertex of the parabola formed by the equation y = (x + 1)2 + 2 is found one unit to the left of the y-axis and two units above the x-axis, at the point (C1, 2). Alternatively, test each answer choice by plugging the x value of the choice into the equation and solving for y. Only the coordinates in choice C, (C1, 2), represent a point on the parabola:
(y = (x + 1)2 + 2
2 = (C1 + 1)2 + 2,
2 = (0)2 + 2,
2 = 2
so it is the only point of the choices given that could be the vertex of the parabola.
is equivalent to:
- A .
- B .
- C .
- D .
- E .
A
Explanation:
When a base is raised to a fractional exponent, raise the base to the power given by the numerator and take the root given by the denominator. Raise the base, a , to the b th power, since b is the numerator of the exponent. Then, take the c th root of that:
.
If the statement “No penguins live at the North Pole” is true, which of the following statements must also be true?
- A . All penguins live at the South Pole.
- B . If Flipper is not a penguin, then he lives at the North Pole.
- C . If Flipper is not a penguin, then he does not live at the North Pole.
- D . If Flipper does not live at the North Pole, then he is a penguin.
- E . If Flipper lives at the North Pole, then he is not a penguin.
E
Explanation:
No penguins live at the North Pole, so anything that lives at the North Pole must not be a penguin. If Flipper lives at the North Pole, then he, like all things at the North Pole, is not a penguin.
If p < 0, q > 0, and r > p, then which of the following must be true?
- A . p + r > 0
- B . rp < rq
- C . pr < rq
- D . r + q > q
- E . p + r < r + q
E
Explanation:
If p < 0, q > 0, then p <q. Since p < q, p plus any value will be less than q plus that same value (whether positive or negative). Therefore, p + r < r + q.
The pie chart above shows the distribution of video rentals from Al’s Video Vault for a single night.
If 250 videos were rented that night, how many more action movies were rented than horror movies?
- A . 10
- B . 20
- C . 22
- D . 25
- E . 30
D
Explanation:
22% of the movies rented were action movies; 250 × 0.2 2 = 55 movies; 12% of the movies rented were horror movies; 250 × 0.1 2 = 30 movies. There were 55 C 30 = 25 more action movies rented than horror movies.
If the circumference of the circle in the diagram above is 20π units, what is the area of triangle ABC?
- A . 40 square units B. 80 square units
- B . 80π square units
- C . 160 square units
- D . 160π square units
B
Explanation:
The circumference of a circle is equal to 2πr, where r is the radius of the circle. If the circumference of the circle = 20π units, then the radius of the circle is equal to ten units. The base of triangle ABC is the diameter of the circle, which is twice the radius. The base of the triangle is 20 units and the height of the triangle is eight units. The area of a triangle is equal to 1/2 bth where b is the base of the triangle and h is the height of the triangle. The area of triangle ABC = 1/2 × 80 × 20 = 1/2 × 160 = 80 square units.
The area of an isosceles right triangle is 18 cm2.What is the length of the hypotenuse of the triangle?
- A . 6 cm
- B .
cm - C .
cm - D .
cm - E .
cm
B
Explanation:
The area of a triangle is equal to 1/2bh where b is the base of the triangle and h is the height of the triangle. The base and height of an isosceles right triangle are equal in length. Therefore, 1/2b 2 = 18, b 2 =36. B=6. The legs of the triangle are 6 cm. The hypotenuse of an isosceles right triangle is equal to the length of one leg multiplied by √2. The hypotenuse of this triangle is equal to 6√2 cm
If 
anda = 4 and b = 8, which of the following could be true?
- A . x < a
- B . x > b
- C . a < x < b
- D . 4 < x < 8
- E . None of the above
A
Explanation:
If a = 4, x could be could be less than a . For example, x could be 3:
,
,
. Although x < a is not true for all values of x , it is true for some values of x .
The length of a rectangle is one greater than three times its width. If the perimeter of the rectangle is 26 feet, what is the area of the rectangle?
- A . 13 ft 2
- B . 24 ft 2
- C . 30 ft 2
- D . 78 ft 2
- E . 100 ft 2
C
Explanation:
The perimeter of a rectangle is equal to 2 l + 2 w , where l is the length of the rectangle and w is the width of the rectangle. If the length is one greater than three times the width, then set the width equal to x and set the length equal to 3 x + 1:
2 × (3 x + 1) + 2 x = 26
6 x + 2 + 2 x = 26
8 x = 24
x = 3
The width of the rectangle is 3 ft and the length of the rectangle is 10 ft. The area of a rectangle is equal to lw ; 10 ft × 3 ft = 30 ft 2 .
Based on the diagram above, which of the following is true?
- A . i = e + f
- B . g + i = h + e
- C . e + i = e + h
- D . e + g + i = 180
- E . e + f + g + h + i = 360
A
Explanation:
The measure of an exterior angle of a triangle is equal to the sum of the two interior angles of the triangle to which the exterior angle is NOT supplementary.
is supplementary to
, so the sum of the interior
and
is
equal to the measure of
.
Which of the following is an irrational number?
- A .
- B . 4 C3
- C .
D
- D .
E
Explanation:
An irrational number is a number that cannot be expressed as a repeating or terminating decimal.
. √2 cannot be expressed as a repeating or terminating decimal, therefore, 128√2 is an irrational number.
In the diagram above, the length of a side of square ABCD is four units.
What is the area of the shaded region?
- A . 4
- B . 4 C π
- C . 4 C 4π
- D . 16π
- E . 16 C 4π
B
Explanation:
The area of a square is equal to S2 , where S is the length of a side of the square. The area of ABCD is 4 2 = 16 square units. The area of a circle is equal to π r2 , where r is the radius of the circle.
The diameter of the circle is four units. The radius of the circle is 4/2 = 2 square units. The area of the circle is equal to π × 2 2 = 4π. The shaded area is equal to one-fourth of the difference between the area of the square and the area of the circle: 1/4 × (16 C 4π) = 4 C π.
The value of d is increased 50%, then decreased 50%.
Compared to its original value, the value of d is now:
- A . 25% smaller.
- B . 25% larger.
- C . 50% smaller.
- D . 50% larger.
- E . the same.
A
Explanation:
To increase d by 50%, multiply d by 1.5: d = 1.5d. To find 50% of 1.5d, multiply 1.5d by 0.5: 1.5d × 0.5 = 0.75d. Compared to its original value, d is now 75% of what it was. The value of d is now 25% smaller.
Which of the following expressions is undefined when x = C2?
- A .
- B .
- C .
- D .
- E .
E
Explanation:
An expression is undefined when a denominator of the expression is equal to zero. When x = C2, x 2 + 6 x + 8 = (C2) 2 + 6 × (C2) + 8 = 4 C 12 + 8 = 0.
If graphed, which of the following pairs of equations would be parallel to each other?
- A . y = 2x + 4, y = x + 4
- B . y = 3x + 3, y = C1/3x C 3
- C . y = 4x + 1, y = 1/5x + 5
- D . y = 5x + 5, y = 1/5x + 5
- E . y = 6x + 6, y = 6x C 4
E
Explanation:
Parallel lines have the same slope. The lines y = 6x + 6 and y = 6x C 6 both have a slope of 6, so they are parallel to each other.
If 
, then when a = 8, b could be equal to:
- A . C2
- B . 4
- C . 6
- D . 7
- E . 8
C
Explanation:
Substitute 8 for a :
. Rewrite 1 as 8/8 and add it to 4 b /8, then cross multiply:
4 b 2 C 8 b C 32 = 64
b 2 C 2 b C 8 = 16
b 2 C 2 b C 24 = 0
( b C 6) × ( b + 4) = 0
b C 6 = 0, b = 6
b + 4 = 0, b + C4
The average of five consecutive odd integers is C21.
What is the least of these integers?
- A . C17
- B . C19
- C . C21
- D . C23
- E . C25
E
Explanation:
If the average of five consecutive odd integers is C21, then the third integer must be C21. The two larger integers are C19 and C17 and the two lesser integers are C23 and C25. C25 is the least of the five integers. Remember, the more a number is negative, the less is its value.
Line AC is a diagonal of square ABCD.
What is the sine of
- A . 1/2
- B .
- C .
- D .
- E . Cannot be determined
C
Explanation:
A square has four right (90-degree) angles. The diagonals of a square bisect its angles. Diagonal AC bisects C, forming two 45-degree angles,
and
. The sine of 45º is equal to
.
If the height of a cylinder is doubled and the radius of the cylinder is halved, the volume of the cylinder:
- A . remains the same.
- B . becomes twice as large.
- C . becomes half as large.
- D . becomes four times larger.
- E . becomes four times smaller.
C
Explanation:
The volume of a cylinder is equal to π r 2 h , where r is the radius of the cylinder and h is the height. The volume of a cylinder with a radius of 1 and a height of 1 is π. If the height is doubled and the radius is halved, then the volume becomes π × (1/2) 2 × 2 × 1 = π × 1/4 ×2 = 1/2π. The volume of the cylinder has become half as large.