Deck 4: Rational Expressions

Question 1

Perform the multiplication. Simplify the answer, if possible. $\frac { 4 s a } { 7 z } \cdot \frac { 7 z a } { 3 s }$

A) $\frac { 2 } { 7 }$
B) $\frac { 4 s } { 3 }$
C) $\frac { 4 } { 3 } a ^ { 2 }$
D) $\frac { 4 } { 7 z }$
E) $\frac { a ^ { 2 } } { 4 }$

Answer 1

$\frac { 4 } { 3 } a ^ { 2 }$

Question 2

Write the expression in lowest terms. If the expression is already in lowest terms, so indicate. Assume that the denominator does not equal zero. $\frac { 9 y + 9 a } { y ^ { 2 } + y a }$

A) $\frac { 9 } { a }$
B) $\frac { 9 } { y + a }$
C) $\frac { 9 } { y }$
D) $\frac { 9 y } { a }$
E)in lowest terms

Answer 2

$\frac { 9 } { y }$

Question 3

Perform the indicated division and express your answer in simplest form. $\frac { 12 s a } { 3 s ^ { 2 } - 3 s a } \div \frac { s a + a } { 2 s - 2 a }$

A) $\frac { 8 ( s + 1 ) } { ( s - a ) ^ { 2 } }$
B) $\frac { 8 } { s + 1 }$
C) $\frac { 5 + a } { 8 }$
D) $\frac { 8 } { s + a }$
E) $\frac { s + 1 } { 8 }$

Answer 3

$\frac { 8 } { s + 1 }$

Question 4

Find the quotient. $\left( - \frac { 3 } { 5 } \right) \div \left( - \frac { 13 } { 10 } \right)$

A) $\frac { 6 } { 65 }$
B) $- \frac { 6 } { 13 }$
C) $\frac { 6 } { 13 }$
D) $- \frac { 2 } { 13 }$
E) $\frac { 2 } { 13 }$

Question 5

Express the rational number in reduced form. $$\frac { 25 } { 30 }$$

A) $\frac { 5 } { 6 }$
B) $\frac { 6 } { 7 }$
C) $6$
D) $\frac { 1 } { 6 }$
E) $\frac { 1 } { 2 }$

Question 6

Perform the multiplication. Simplify the answer, if possible. $\frac { 2 s } { 5 y } \cdot \frac { 5 y } { 3 s }$

A) $\frac { 2 s } { 3 y }$
B) $\frac { 2 } { 5 y }$
C) $\frac { 8 } { 7 }$
D) $\frac { 2 s } { 3 }$
E) $\frac { 5 } { 3 y }$

Question 7

Simplify the fraction to lowest terms. $\frac { 3,500 } { 1,500 }$

A) $\frac { 7 } { 3 }$
B) $\frac { 70 } { 30 }$
C) $\frac { 3,500 } { 1,500 }$
D) $\frac { 35 } { 15 }$
E) $\frac { 3 } { 7 }$

Question 8

Multiply and reduce your answer to lowest terms. $- \frac { 42 } { 5 } \cdot \frac { 7 } { 21 } \cdot \frac { 1 } { 14 }$

A) $- \frac { 1 } { 5 }$
B) $\frac { 1 } { 5 }$
C)-1
D)1
E) $- \frac { 5 } { 5 }$

Question 9

Simplify the rational expression. $\frac { 8 x ^ { 2 } y + 24 x y ^ { 2 } - 32 y ^ { 3 } } { 48 x ^ { 3 } y - 36 x ^ { 2 } y ^ { 2 } - 12 x y ^ { 3 } }$

A) $\frac { 2 ( x - 4 y ) } { 3 x ( 4 x + y ) }$
B) $\frac { 2 y ( x + 4 y ) } { 3 x ( x - 4 y ) }$
C) $\frac { 2 } { 3 x }$
D) $\frac { 2 ( x + 4 y ) } { 3 x ( 4 x + y ) }$
E) $\frac { 2 ( x - 4 y ) } { 3 x y ( 4 x - y ) }$

Question 10

Perform the multiplication. Simplify the result when possible. $\frac { 5 } { 63 } \cdot \frac { 72 } { 5 }$

A) $\frac { 10 } { 9 }$
B) $\frac { 8 } { 7 }$
C) $\frac { 10 } { 11 }$
D) $\frac { 11 } { 10 }$
E) 9
$$8$$

Question 11

Perform the indicated division. $\frac { ( 2 x - 1 ) ^ { 2 } } { 200 x - 100 } \div \frac { 10 x ^ { 2 } + 3 x - 4 } { 10 x + 5 }$

A) $\frac { 1 } { 2 }$
B)2
C) $\frac { 2 x + 1 } { 4 x + 8 }$
D) $\frac { 2 x + 1 } { 10 x + 8 }$
E) $\frac { 10 x + 8 } { 2 x + 1 }$

Question 12

Find the quotient. $\frac { 1 } { 2 } \div \frac { 5 } { 3 }$

A) $\frac { 2 } { 7 }$
B) $\frac { 3 } { 10 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 7 } { 13 }$
E) $\frac { 8 } { 11 }$

Question 13

Perform the operations. $\frac { 2 s ^ { 2 } + 12 s - 80 } { s ^ { 2 } - 16 } \cdot \frac { s - 2 } { s ^ { 2 } - 100 } \div \frac { s ^ { 2 } - 4 } { s ^ { 2 } + 6 s + 8 }$

A) $\frac { 2 } { s - 10 }$
B) $\frac { 2 } { s + 4 }$
C) $\frac { 2 } { s + 10 }$
D) $\frac { 2 } { s - 20 }$
E) $\frac { 1 } { s - 10 }$

Question 14

Perform the operation. $\frac { 13 } { 3 } \left( - \frac { 18 } { 13 } \right)$

A) $13$
B) $- 6$
C) $- 13$
D) $6$
E)none of these

Question 15

Perform the indicated division and express your answer in simplest form. $\frac { 2 y ^ { 2 } - 3 y - 20 } { ( y - 4 ) ^ { 2 } } \div \frac { 2 y ^ { 2 } + 13 y + 20 } { 2 y ^ { 2 } - 7 y - 4 }$

A) $\frac { y - 4 } { 2 y + 1 }$
B) $\frac { y + 4 } { 2 y + 1 }$
C) $\frac { 2 y + 1 } { y - 4 }$
D) $\frac { 2 y - 1 } { y + 4 }$
E) $\frac { 2 y + 1 } { y + 4 }$

Question 16

Write the expression in lowest terms. If the expression is already in lowest terms, so indicate. Assume that the denominator does not equal zero. $\frac { x + 7 } { 9 ( x + 7 ) }$

A) $\frac { x } { 7 }$
B) $\frac { 1 } { 9 }$
C) $\frac { 9 } { x + 7 }$
D) $\frac { x + 7 } { 9 }$
E)in lowest terms

Question 17

Perform the multiplication. Simplify the answer, if possible. $\frac { 2 z - 4 } { z + 2 } \cdot \frac { 5 } { 5 z - 10 }$

A) $\frac { 2 } { z - 2 }$
B) $\frac { 5 } { z - 2 }$
C) $\frac { z - 2 } { 5 }$
D) $\frac { 5 } { z + 2 }$
E) $\frac { 2 } { z + 2 }$

Question 18

Perform the multiplication. Simplify the answer, if possible. $\frac { t z ^ { 2 } } { t ^ { 2 } z } \cdot \frac { 9 z ^ { 2 } a ^ { 2 } } { t z a } \div \frac { t z a ^ { 2 } } { t ^ { 3 } a ^ { 2 } }$

A) $\frac { 9 z ^ { 2 } a } { t ^ { 3 } }$
B) $\frac { 2 t ^ { 3 } a } { z ^ { 4 } }$
C) $2 z ^ { 3 } a$
D) $\frac { 9 z ^ { 3 } a } { t ^ { 4 } }$
E) $9 z a$

Question 19

Find the quotient. $\frac { 5 } { 2 } \div \left( - \frac { 3 } { 7 } \right)$

A) $\frac { 40 } { 5 }$
B) $- \frac { 36 } { 11 }$
C) $- \frac { 39 } { 3 }$
D) $- \frac { 34 } { 10 }$
E) $- \frac { 35 } { 6 }$

Question 20

Express the rational number in reduced form. $- \frac { 40 } { 50 }$

A) $\frac { 1 } { 2 }$
B) $- \frac { 4 } { 5 }$
C) $\frac { 1 } { 5 }$
D) $\frac { 4 } { 5 }$
E) $- \frac { 5 } { 6 }$

Question 21

Perform the indicated operation and express your answer in simplest form. $\frac { 6 } { x ^ { 2 } - 5 x - 24 } - \frac { 1 } { x ^ { 2 } - 3 x - 40 }$

A) $\frac { x + 27 } { ( x - 8 ) ( x - 3 ) ( x - 5 ) }$
B) $\frac { 5 x + 27 } { ( x - 8 ) ( x + 3 ) ( x - 5 ) }$
C) $\frac { 5 x + 27 } { ( x - 8 ) ( x + 3 ) ( x + 5 ) }$
D) $\frac { 5 x + 27 } { ( x - 8 ) ( x - 3 ) ( x + 5 ) }$
E) $\frac { x + 5 } { ( x - 8 ) ( x - 3 ) ( x - 5 ) }$

Question 22

Simplify the complex fraction. $\frac { \frac { 5 } { x } - \frac { 6 } { y } } { \frac { 7 } { y } - \frac { 2 } { x y } }$

A) $\frac { x - 6 y } { 7 x - 2 }$
B) $\frac { 5 y - 6 x } { 7 x - 2 }$
C) $\frac { x - 7 y } { x y ^ { 2 } }$
D) $\frac { x - 7 y } { x y }$
E) $\frac { 5 x - 6 y } { 7 - 2 x }$

Question 23

Perform the division. $\frac { 12 x - 3 y + 36 z } { 12 x y }$

A) $\frac { 3 } { 4 } + \frac { 3 z } { x y }$
B) $\frac { 1 } { y } + \frac { 1 } { 4 x } + \frac { 1 } { x y }$
C) $\frac { 1 } { y } - \frac { 1 } { 4 x } + \frac { 3 } { x y }$
D) $\frac { 1 } { y } - \frac { 1 } { 4 x } + \frac { 3 z } { x y }$
E) $\frac { 4 } { y } - \frac { 4 } { 5 x } + \frac { 3 z } { x y }$

Question 24

Use synthetic division to perform the division.

Question 25

Solve the equation. $\frac { 5 n } { 6 } - \frac { n } { 8 } = \frac { - 102 } { 24 }$

A)n = -6
B)n = -12
C)n = 6
D)n = 8
E)n = -9

Question 26

Perform the division. $\frac { - 10 a ^ { 2 } b ^ { 2 } - 20 a ^ { 2 } b - 15 a b ^ { 2 } } { - 5 a b }$

A) $3 a b + 5 a + 3 b$
B) $3 a b + 4 a + 3 b$
C) $2 a b + 5 a + 3 b$
D) $2 a b + 4 a + 4 b$
E) $2 a b + 4 a + 3 b$

Question 27

Do the addition. Simplify the result when possible. $\frac { 1 } { 9 } + \frac { 1 } { 27 }$

A) $\frac { 4 } { 3 }$
B) $\frac { 2 } { 27 }$
C) $\frac { 4 } { 27 }$
D) $\frac { 4 } { 9 }$
E) $\frac { 7 } { 27 }$

Question 28

Simplify the complex fraction. $\frac { - \frac { 7 } { x } - \frac { 14 } { x + 7 } } { \frac { 2 } { x ^ { 2 } + 7 x } + \frac { 2 } { x } }$

A) $- \frac { x + 7 } { 2 ( x + 8 ) }$
B) $- \frac { 3 x - 7 } { 7 ( x + 8 ) }$
C) $- \frac { 7 ( 3 x - 7 ) } { 2 ( x + 8 ) }$
D) $- \frac { 2 ( 3 x - 7 ) } { 7 ( x + 8 ) }$
E) $- \frac { 7 ( 3 x - 7 ) } { 2 ( x + 8 ) }$ .

Question 29

Perform the division. $\frac { 16 a ^ { 7 } b ^ { 6 } - 24 a ^ { 6 } b ^ { 7 } } { 4 a ^ { 6 } b ^ { 6 } }$

A) $4 b - 6 a$
B) $2 a - 2 b$
C) $4 a - 6 b$
D) $4 b - 7 a$
E) $2 a ^ { 2 } - 2 b$

Question 30

Perform the operation involving rational numbers. Be sure to express your answer in reduced form. $\frac { 1 } { 3 } + \frac { 5 } { 6 } - \frac { 4 } { 9 }$

A) $\frac { 29 } { 18 }$
B) $\frac { 13 } { 9 }$
C) $\frac { 13 } { 17 }$
D) $\frac { 13 } { 18 }$
E) $\frac { 18 } { 13 }$

Question 31

Use a calculator and synthetic division to perform the division. $\left( 15 x ^ { 3 } - 25 \right) \div ( x + 53 )$

A) $15 x ^ { 2 } - 795 x + 42,135 - \frac { 2,233,180 } { x + 53 }$
B) $15 x ^ { 2 } - 795 x + 42,180 - \frac { 2,233,250 } { x + 53 }$
C) $15 x ^ { 2 } - 795 x + 42,160 - \frac { 2,233,250 } { x + 53 }$
D) $15 x ^ { 2 } - 795 x + 42,170 - \frac { 2,233,240 } { x + 53 }$

Question 32

Simplify the complex fraction. $\frac { \frac { 1 } { 2 } - \frac { 1 } { 4 } } { \frac { 5 } { 6 } + \frac { 3 } { 4 } }$

A) $\frac { 1 } { 21 }$
B) $\frac { 3 } { 21 }$
C) $\frac { 1 } { 19 }$
D) $\frac { 2 } { 19 }$
E) $\frac { 3 } { 19 }$

Question 33

Add the rational expressions. Be sure to express your answer in simplest form. $\frac { x - 5 } { 6 } + \frac { x + 7 } { 7 }$

A) $\frac { 36 } { 13 x + 7 }$
B) $\frac { 13 x - 77 } { 42 }$
C) $\frac { 13 x + 7 } { 42 }$
D) $\frac { 13 x - 7 } { 42 }$
E) $\frac { 2 x + 7 } { 13 }$

Question 34

Use synthetic division to perform the division. $\left( x ^ { 2 } - 12 x + 32 \right) \div ( x - 4 )$

A)x + 10
B)x + 7
C)x - 7
D)x - 8
E)x - 10

Question 35

Perform the indicated operations involving rational numbers. Express the final answer in reduced form. $\frac { 4 } { 9 } \cdot \frac { 6 } { 13 } \div \frac { 2 } { 9 }$

A) $1$
B) $\frac { 1 } { 13 }$
C) $\frac { 11 } { 12 }$
D) $\frac { 1 } { 12 }$
E) $\frac { 12 } { 13 }$

Question 36

Perform the operation. $\frac { 1 } { 2 } - \left( - \frac { 3 } { 5 } \right)$

A) $\frac { 11 } { 10 }$
B) $\frac { 3 } { 10 }$
C) $\frac { 1 } { 10 }$
D) $\frac { 1 } { 5 }$
E) $- \frac { 1 } { 10 }$

Question 37

Solve the equation. $\frac { y } { 4 } + \frac { y - 68 } { 4 } = \frac { 3 y + 6 } { 2 }$

A)y = -4
B)y = -5
C)y = -19
D)y = -22
E)y = -20

Question 38

Perform the indicated operation and express your answer in simplest form. $\frac { 11 x + 41 } { x ^ { 2 } - 9 } - \frac { 6 } { x - 3 }$

A) $\frac { 11 x + 23 } { ( x + 3 ) ( x - 3 ) }$
B) $\frac { 5 x - 23 } { ( x + 3 ) ( x - 3 ) }$
C) $\frac { 5 x - 23 } { x + 3 }$
D) $\frac { 5 x + 23 } { ( x + 3 ) ( x - 3 ) }$
E) $\frac { 5 x - 23 } { x - 3 }$

Question 39

Perform the indicated operation and express your answer in simplest form. $\frac { 5 x } { x ^ { 2 } + 8 x } + \frac { 6 } { x }$

A) $\frac { x + 48 } { x + 8 }$
B) $\frac { 5 x + 48 } { x ( x + 8 ) }$
C) $\frac { 11 x + 48 } { x + 8 }$
D) $\frac { 11 x + 48 } { x + 2 }$
E) $\frac { 11 x + 48 } { x ( x + 8 ) }$

Question 40

Add or subtract the rational expressions. Be sure to express your answer in simplest form. $\frac { 5 } { 4 x } + \frac { 6 } { 5 y } - 2$

A) $\frac { 20 y + 30 x - 8 x - 10 y } { 4 x + 5 y }$
B) $\frac { 25 y + 24 x - 40 x y } { 20 x y }$
C) $\frac { 25 y + 24 x } { 4 x + 5 y } - 2$
D) $\frac { 25 y + 24 x } { 20 x y }$
E) $\frac { 25 y + 24 x - x y } { 20 x y }$

Question 41

Solve the equation. $$3 + \frac { 12 } { t - 7 } = \frac { 12 } { t ( t - 7 ) }$$

A){3, -1 }
B){ -1}
C){3}
D){4, -1}
E){-3, 1}

Question 42

Solve the proportion for the variable. $\frac { 6 } { y } = \frac { 9 } { 30 }$

A)22
B)23
C)16
D)25
E)20

Question 43

To travel 68 miles, it takes Sue, riding a moped, 4 hours less time than it takes Doreen to travel 80 miles riding a bicycle. Sue travels 7 miles per hour faster than Doreen. Find the times and rates of both girls.

A)Sue: 17 mph for 4 hours, Doreen: 10 mph for 8 hours
B)Sue: 10 mph for 4 hours, Doreen: 17 mph for 8 hours
C)Sue: 21 mph for 8 hours, Doreen: 14 mph for 12 hours
D)Sue: 4 mph for 17 hours, Doreen: 10 mph for 8 hours
E)Sue: 27 mph for 4 hours, Doreen: 20 mph for 8 hours

Question 44

Use similar triangles to help solve the problem. A tree casts a shadow of 24 feet at the same time as a 5-foot tall man casts a shadow of 4 feet. Find the height of the tree.

A)h = 27 ft
B)h = 42 ft
C)h = 30 ft
D)h = 35 ft
E)h = 22 ft

Question 45

Solve the equation. $$\frac { 2 y - 10 } { 8 y ^ { 2 } + 18 y - 18 } - \frac { 4 } { 2 y + 6 } = \frac { 4 } { 4 y - 3 }$$

A){ 0, 1 }
B){ 0, -1 }
C){ 0 }
D){ -1 }
E){ 1 }

Question 46

A school board has determined that there should be 3 teachers for every 50 students. How many teachers are needed for an enrollment of 2,650 students?

A)134 teachers
B)148 teachers
C)159 teachers
D)143 teachers
E)216 teachers

Question 47

Use similar triangles to help solve the problem. An airplane ascends 160 feet as it flies a horizontal distance of 1,000 feet. How much altitude will it gain as it flies a horizontal distance of 1 mile? ( Hint: 5,280 feet = 1 mile.)Round your answer to the nearest integer.

A)x = 870 ft
B)x = 813 ft
C)x = 845 ft
D)x = 885 ft
E)x = 833 ft

Question 48

An inlet pipe can fill a tank in 8 minutes. A drain can empty the tank in 10 minutes. If the tank is empty and both the pipe and drain are open, how long will it take before the tank overflows?

A)38 minutes
B)20 minutes
C)2 minutes
D)50 minutes
E)40 minutes

Question 49

Use similar triangles to help solve the problem. A man places a mirror on the ground and sees the reflection of the top of a flagpole as shown in the figure below. The two triangles in the figure are similar. Find the height h of the flagpole. a = 11, b = 7, c = 31

A) $14$ ft
B) $21$ ft
C) $19 \frac { 8 } { 11 } \mathrm { ft }$
D) $22 \frac { 8 } { 11 } \mathrm { ft }$

Question 50

Solve the following equation for the variable x . $y = - \frac { l } { g } x + \frac { e } { n }$

A) $x = \frac { g e - g n y } { \lg }$
B) $x = \frac { l n + g n y } { e n }$
C) $x = \frac { g e - g n y } { \ln }$
D) $x = y \left( \frac { g } { l } - \frac { n } { e } \right)$
E) $x = y \left( \frac { g e - g n } { \ln } \right)$

Question 51

Solve the following equation for the variable y . $- \frac { 2 } { x - 4 } = \frac { 3 } { y - 1 }$

A) $y = - \frac { 2 } { 3 } x + 7$
B) $y = \frac { 3 } { 2 } x + 7$
C) $y = \frac { 2 } { 3 } x - 7$
D) $y = - \frac { 2 } { 3 } x - 7$
E) $y = - \frac { 3 } { 2 } x + 7$

Question 52

Solve the equation. $\frac { 3 a - 9 } { 6 } + \frac { 4 a - 4 } { 4 } + \frac { 3 a + 9 } { 4 } = 11$

A)a = 10
B)a = -5
C)a = 2
D)a = 5
E)a = 8

Question 53

Kent drives his Mazda 270 miles in the same time that Dave drives his Nissan 250 miles. If Kent averages 4 miles per hour faster than Dave, find their rates.

A)46 mph for Dave and 50 mph for Kent
B)45 mph for Dave and 49 mph for Kent
C)60 mph for Dave and 64 mph for Kent
D)50 mph for Dave and 54 mph for Kent
E)54 mph for Dave and 46 mph for Kent

Question 54

Solve the equation. $$5 + \frac { 25 } { n - 5 } = \frac { 125 } { n ^ { 2 } - 5 n }$$

A){1}
B){0, 9}
C){3}
D){0, 3}
E){-3}

Question 55

Solve the following equation for the variable x . $$y = \frac { 4 } { 5 } x + \frac { 5 } { 20 }$$

A) $$x = - \frac { 5 } { 4 } y + \frac { 5 } { 16 }$$
B) $x = \frac { 5 } { 4 } y + \frac { 5 } { 4 }$
C) $x = \frac { 5 } { 4 } y - \frac { 5 } { 16 }$
D) $x = \frac { 5 } { 4 } y - \frac { 5 } { 20 }$
E) $x = \frac { 5 } { 4 } y + \frac { 5 } { 20 }$

Question 56

Solve for the variable in the proportion. $\frac { x + 2 } { 4 } = \frac { 6 } { 12 }$

A) $x = 1$
B) $x = - 2$
C) $x = 3$
D) $x = - 1$
E) $x = 0$