Deck 7: Rational Expressions and Equations

Question 1

Perform the indicated operation and simplify. $$\frac { p ^ { 3 } r ^ { 2 } } { c d } \cdot c ^ { - 4 } d ^ { 3 }$$

A) $\frac { p ^ { 3 } r ^ { 2 } d } { c ^ { 5 } }$
B) $\frac { p ^ { 3 } r ^ { 2 } d ^ { 2 } } { c ^ { 5 } }$
C) $\frac { c r ^ { 2 } d } { p ^ { 5 } }$
D) $\frac { p ^ { 2 } r d ^ { 5 } } { d ^ { 2 } }$
E) $\frac { p r ^ { 2 } d ^ { 2 } } { c ^ { 5 } }$

Answer 1

$\frac { p ^ { 3 } r ^ { 2 } d ^ { 2 } } { c ^ { 5 } }$

Question 2

Perform the indicated division and express your answer in simplest form.

Answer 2

Question 3

Perform the indicated operation and simplify. $$\frac { z + 16 } { v ^ { 2 } + 22 v + 96 } \cdot ( v + 6 )$$

A) $( z + 16 ) \cdot ( v + 16 )$
B) $\frac { v + 6 } { v + 16 }$
C) $\frac { z + 16 } { v + 6 }$
D) $\frac { z + 16 } { v + 16 }$
E) $\frac { z + 6 } { v + 6 }$

Answer 3

$\frac { z + 16 } { v + 16 }$

Question 4

Solve the expression. $$\frac { 18 x } { y } \cdot \frac { z } { 19 w } = \frac { ? } { ? } ( y \neq 0 , w \neq 0 )$$

A) $\frac { 19 } { 18 } \cdot \frac { x y } { z w }$
B) $\frac { 18 } { 19 } \cdot \frac { x z } { y w }$
C) $\frac { 19 } { 18 } \cdot \frac { x w } { y z }$
D) $\frac { 18 } { 19 } \cdot \frac { x w } { y z }$

Question 5

Which value of x makes the rational expression $\frac { x ^ { 2 } - 5 x } { x ^ { 2 } + 9 }$ undefined?

A) $3$
B) $- 3$
C) $0$
D) $5$
E)none

Question 6

Write the expression in lowest terms. If the expression is already in lowest terms, so indicate. Assume that the denominator does not equal zero. $$\frac { x + 3 } { 9 ( x + 3 ) }$$

A) $\frac { x } { 3 }$
B) $\frac { 1 } { 9 }$
C) $\frac { 9 } { x + 3 }$
D) $\frac { x + 3 } { 9 }$
E)in lowest terms

Question 7

Perform the indicated division and express your answer in simplest form. $$\frac { 4 s ^ { 2 } + 3 s r - r ^ { 2 } } { s ^ { 2 } r } \div \frac { 5 s ^ { 2 } + 4 s r - r ^ { 2 } } { r }$$

A) $\frac { 5 s + r } { s ^ { 2 } ( 4 s + r ) }$
B) $\frac { 4 s + r } { s ^ { 2 } ( 5 s - r ) }$
C) $\frac { 5 s - r } { s ^ { 2 } ( 4 s - r ) }$
D) $\frac { 4 s - r } { s ^ { 2 } ( 5 s - r ) }$
E) $\frac { 4 s - r } { s ^ { 2 } ( 5 s + r ) }$

Question 8

Simplify the rational expression. $$\frac { 16 x ^ { 2 } y + 32 x y ^ { 2 } - 48 y ^ { 3 } } { 60 x ^ { 3 } y - 40 x ^ { 2 } y ^ { 2 } - 20 x y ^ { 3 } }$$

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

Question 9

Perform the indicated operation and simplify. $$\frac { x ^ { 2 } - 9 } { x ^ { 2 } - 81 } \div \frac { x + 3 } { x + 9 }$$

A) $\frac { x + 11 } { x - 9 }$
B) $\frac { x - 3 } { x - 9 }$
C) $x ^ { 2 } + 5$
D) $\frac { x - 9 } { x - 3 }$
E) $\frac { x + 3 } { x + 9 }$

Question 10

Find any real numbers x which make $\frac { x - 2 } { x ^ { 2 } - 3 x - 10 }$ undefined.

A) $x = - 2 , x = 5$
B) $x = 2 , \quad x = - 5$
C) $x = 2$
D)defined for all real numbers

Question 11

Simplify this rational expression. Assume the denominator never equals zero. $\frac { 3 - 3 \cdot y } { y ^ { 2 } - 1 }$

A) $- \frac { 3 } { y + 1 }$
B) $\frac { 1 } { y - 1 }$
C) $\frac { 3 } { y + 1 }$
D) $\frac { 3 } { y ^ { 2 } - 1 }$
E)Does not simplify further.

Question 12

Perform the indicated operation and simplify. $$\frac { 3 b ^ { 2 } - 14 b - 5 } { b ^ { 2 } - 25 } \cdot \frac { 2 b ^ { 2 } + 8 b - 10 } { 3 b ^ { 2 } + 10 b + 3 }$$

A) $\frac { b - 2 } { b + 2 }$
B) $\frac { 2 b } { b + 6 }$
C) $\frac { 5 b + 2 } { b + 6 }$
D) $\frac { 2 b - 2 } { b + 3 }$
E) $\frac { 5 b - 3 } { b - 3 }$

Question 13

Simplify this rational expression. Assume the denominator never equals zero. $$\frac { 8 x - 2 a } { 2 a - 8 x }$$

A)1
B)2 a - 8 x
C)8 x - 2 a
D)-1
E)Does not simplify further.

Question 14

Write the expression in lowest terms. If the expression is already in lowest terms, so indicate. Assume that the denominator does not equal zero. $$\frac { 8 k + 8 c } { k ^ { 2 } + k c }$$

A) $\frac { 8 } { c }$
B) $\frac { 8 } { k + c }$
C) $\frac { 8 } { k }$
D) $\frac { 8 k } { c }$
E)in lowest terms

Question 15

Perform the indicated operation and simplify. $$\frac { 6 } { z } \cdot z ^ { 3 }$$

A) $\frac { z ^ { 1 } } { 6 }$
B) $\frac { 6 } { z ^ { 4 } }$
C) $6 z ^ { 2 }$
D) $6 z ^ { 4 }$

Question 16

Evaluate the following expression for the given value of the variable. $\frac { x - 8 } { 4 }$ for $x = 28$

A) $-$ 20
B)13
C)20
D)5
E) $-$ 5

Question 17

Evaluate: $\frac { - x - 8 } { x ^ { 2 } - 4 x - 12 } \text { for } x = - 4$

A) $- \frac { 3 } { 5 }$
B) $- \frac { 1 } { 5 }$
C) $\frac { 1 } { 5 }$
D) $- \frac { 1 } { 2 }$

Question 18

Perform the division. $$\left( 3 y - 12 + 15 y ^ { 2 } \right) \div ( 4 - 5 y )$$

A) $3 - y$
B) $- 3 y - 3$
C) $- y - 3$
D) $3 y - 3$
E) $y - 3$

Question 19

Perform the indicated operation and simplify. $$\frac { p ^ { 5 } r ^ { 3 } } { c d } \cdot \frac { c ^ { - 4 } d ^ { 5 } } { p }$$

A) $\frac { p ^ { 2 } r ^ { 3 } d } { c ^ { 5 } }$
B) $\frac { p ^ { 4 } r ^ { 3 } d ^ { 4 } } { c ^ { 5 } }$
C) $\frac { c r ^ { 3 } d } { p ^ { 5 } }$
D) $\frac { p ^ { 4 } r ^ { 3 } d ^ { 5 } } { d ^ { 4 } }$
E) $\frac { p r ^ { 3 } d ^ { 3 } } { c ^ { 5 } }$

Question 20

Evaluate the following expression for the given value of the variable. $\frac { x - 6 } { 6 - x }$ for $x = 1$

A)1
B)-5
C) $-$ 1
D)5
E)0

Question 21

Perform the indicated operation.

Question 22

Perform the indicated operation. Simplify the answer if possible. $$\frac { 5 } { a + 9 } - \frac { 2 } { - 9 - a }$$

A) $\frac { 7 } { a - 9 }$
B) $\frac { 7 } { a + 9 }$
C) $\frac { - 7 } { a + 9 }$
D) $\frac { 7 a } { a - 9 }$
E)none of these

Question 23

Perform the indicated operation. Simplify the answer if possible.

Question 24

Perform the operation. Simplify the answer, if possible.

Question 25

Simplify the complex fraction. $$\frac { x ^ { - 8 } + 1 } { x ^ { - 8 } - 1 }$$

A) $\frac { 1 - x ^ { 8 } } { 1 + x ^ { 8 } }$
B) $\frac { x ^ { 8 } } { 1 - x ^ { 8 } }$
C) $\frac { 1 + x ^ { 8 } } { x ^ { 8 } }$
D) $\frac { 1 + x ^ { 8 } } { 1 - x ^ { 8 } }$
E)none of these

Question 26

Add the rational expressions. Be sure to express your answer in simplest form. $$\frac { 3 x } { x - 6 } + \frac { 12 } { x - 6 }$$

A) $\frac { 3 ( x + 4 ) } { x - 6 }$
B) $\frac { 3 ( x - 4 ) } { x - 6 }$
C) $\frac { 3 x + 5 } { x - 6 }$
D) $\frac { 3 x + 4 } { x - 6 }$
E) $\frac { 3 ( x - 4 ) } { x + 6 }$

Question 27

Perform the indicated operation. Simplify the answer if possible. $$\frac { t + 6 } { t - 3 } - \frac { t + 10 } { 3 - t }$$

A) $\frac { 2 t + 16 } { t - 3 }$
B) $\frac { 2 t + 16 } { t + 3 }$
C) $\frac { 2 t - 16 } { t + 3 }$
D) $\frac { 2 t - 16 } { t - 3 }$
E)none of these

Question 28

Simplify the complex fraction: $$\frac { \frac { 14 } { 2 } } { \frac { 10 } { 6 } }$$

A) $\frac { 3 } { 35 }$
B) $\frac { 35 } { 3 }$
C) $\frac { 21 } { 5 }$
D) $\frac { 5 } { 21 }$
E)none of these

Question 29

Perform the indicated operation and express your answer in simplest form. $$\frac { 3 x } { x ^ { 2 } + 8 x } - \frac { 2 } { x }$$

A) $\frac { 3 x - 16 } { x + 8 }$
B) $\frac { 3 x + 16 } { x + 8 }$
C) $\frac { - 3 x + 16 } { x ( x + 8 ) }$
D) $\frac { 3 x - 16 } { x ( x + 8 ) }$
E) $\frac { x - 16 } { x ( x + 8 ) }$

Question 30

Change the fraction $\frac { 4 y } { p }$ into an equivalent fraction with the denominator p ² + p .

A) $\frac { 4 y ( p + 1 ) } { p + 1 }$
B) $\frac { 4 y \left( p ^ { 2 } + p \right) } { p + 1 }$
C) $\frac { 4 y ( p + 1 ) } { p ^ { 2 } + p }$
D) $\frac { 4 y \left( p ^ { 2 } + p \right) } { p ^ { 2 } + p }$
E)none of these.

Question 31

Build the fraction into an equivalent fraction with the indicated denominator.

Question 32

Simplify the complex fraction. $$\frac { - \frac { 2 } { x } - \frac { 4 } { x + 2 } } { \frac { 7 } { x ^ { 2 } + 2 x } + \frac { 7 } { x } }$$

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

Question 33

A jaguar can run 54 mph. How fast is this in kilometers per hour? Round your answer to the nearest tenth. __________ km\h

Question 34

Two denominators are given. Find the LCD. t ² 7 t 8, t ² 64

Question 35

Convert 98 m³ to liters.

A)0.098
B)98000
C)98
D)0.98

Question 36

Add the rational expressions. Be sure to express your answer in simplest form. $$\frac { x - 4 } { 6 } + \frac { x + 8 } { 7 }$$

A) $\frac { 36 } { 13 x + 20 }$
B) $\frac { 13 x - 76 } { 42 }$
C) $\frac { 13 x + 20 } { 42 }$
D) $\frac { 13 x - 20 } { 42 }$
E) $\frac { 2 x + 20 } { 13 }$

Question 37

Perform the indicated operation. Simplify the answer if possible.

Question 38

Convert 6 meters per minute to kilometers per hour.

A)100 kilometers per hour
B)0.36 kilometers per hour
C)6 kilometers per hour
D)66 kilometers per hour

Question 39

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

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

Question 40

Simplify the complex fraction. $$\frac { \frac { 7 } { 6 x } } { \frac { 14 } { 3 x y } }$$

A) $\frac { y } { 2 }$
B) $\frac { x y } { 4 }$
C) $\frac { x } { 2 y }$
D) $\frac { x } { 4 y }$
E) $\frac { y } { 4 }$

Question 41

Solve $d = \frac { 4 c b } { c - b }$ for c .

A) $c = \frac { b d } { 4 b - 1 }$
B) $c = \frac { 1 } { 4 }$
C) $c = \frac { d ( c - b ) } { 4 b }$
D) $c = - \frac { b d } { 4 b - d }$

Question 42

Write a fraction in simplified form that expresses the ratio of 0.24 to 0.32.

A) $\frac { 3 } { 40 }$
B) $\frac { 4 } { 6 }$
C) $\frac { 4 } { 5 }$
D) $\frac { 4 } { 3 }$
E) $\frac { 3 } { 4 }$

Question 43

It takes a grounds keeper 45 minutes to prepare a softball field for a game. It takes his assistant 65 minutes to prepare the same field. How long will it take if they work together to prepare the field?

A) $24 \frac { 13 } { 22 } \mathrm {~min}$
B) $26 \frac { 13 } { 21 } \mathrm {~min}$
C) $25 \frac { 13 } { 22 } \mathrm {~min}$
D) $26 \frac { 13 } { 22 } \mathrm {~min}$
E) $26 \frac { 15 } { 22 } \mathrm {~min}$

Question 44

If a number is added to the numerator of $\frac { 3 } { 8 }$ and twice as much is added to the denominator, the result is $\frac { 4 } { 9 }$ . Find the number.

A)4
B)2
C)5
D)1
E)3

Question 45

A homeowner estimates that it will take her 8 days to roof her house. A professional roofer estimates that he could roof the house in 5 days. How long will it take if the homeowner helps the roofer?

A) $3 \frac { 1 } { 13 } \text { days }$
B) $1 \frac { 1 } { 13 } \text { days }$
C) $2 \frac { 1 } { 13 } \text { days }$
D) $$3 \frac { 3 } { 13 }$$
days
E) $3 \frac { 1 } { 14 } \text { days }$

Question 46

A dealer bought some cell phones for a total of $1,066. He gave away 1 cell phone as a gift, sold the rest for $15 more then he paid for each cell phone, and earned $98. How many cell phones did he buy?

A)14
B)12
C)15
D)13
E)11

Question 47

A man roller-blades at a rate 5 miles per hour faster than he jogs. In the same time it takes him to roller-blade 6 miles he can jog 3 miles. How fast does he jog?

A)4 mph
B)3 mph
C)5 mph
D)1 mph
E)2 mph

Question 48

Solve the equation and check the solution. $$\frac { 3 x - 1 } { 6 } - \frac { x + 3 } { 2 } = \frac { 3 x + 1 } { 3 }$$

A)- 2
B)4
C)3
D)- 3
E)no solution

Question 49

Two certificates of deposit (CDs)pay interest at rates that differ by 4%. Money invested for 1 year in the first CD earns $90 interest. The same principal invested in the second CD earns $150. Find the two rates of interest.

A)6%, 7%
B)5%, 10%
C)6%, 10%
D)4%, 10%
E)6%, 6%

Question 50

Mary invested same amount of money in two bond funds for 1 year. They pay interest at rates that differ by 4%. Money invested in the first fund earns $360, and money invested in the second fund earns $600. Find the bigger rate of interest.

A)6%
B)7%
C)11%
D)12%
E)10%

Question 51

The first bird can fly about 20 mph faster than the second one. In the same time that the first bird travels 120 miles, the second bird travels 60 miles. Find their flying speeds.

A)The first bird: 41 mph; the second bird: 21 mph
B)The first bird: 37 mph; the second bird: 17 mph
C)The first bird: 40 mph; the second bird: 20 mph
D)The first bird: 42 mph; the second bird: 22 mph
E)The first bird: 36 mph; the second bird: 16 mph

Question 52

Solve the equation and check the solution. $$\frac { 3 a } { 4 } + \frac { a } { 2 } = - 20$$

A)- 16
B)- 8
C)16
D)8
E)no solution

Question 53

Solve the formula for t :

Question 54

If the same number is added to both the numerator and the denominator of the amount of pecans used in the following recipe for toffee, the result is the amount of brown sugar to be used. Find the number. $\text { Toffee }$

$\frac{2}{3} \text { cup pecans. chopped }$

$\frac{3}{4} \text { cup brown sugar (packed) }$

$\frac{1}{2} \text { package }(6-0 z \text { size) semisw eet chocolate }$

$\frac{1}{4} \text { cup butter }$

A)1
B)3
C)2
D)5
E)4

Question 55

Write the ratio as a fraction in simplest form. $3$ inches to $12$ inches

A) $\frac { 4 } { 4 }$
B) $\frac { 1 } { 4 }$
C) $4$
D) $\frac { 4 } { 3 }$
E) $\frac { 3 } { 4 }$

Question 56

It takes the owner\operator of a dog kennel 2 hr to clean all of the cages. It takes his assistant 1 hr more than that to clean the same cages. How long will it take if they work together?

A) $2 \frac { 1 } { 5 } \mathrm { hr }$
B) $1 \frac { 3 } { 5 } \mathrm { hr }$
C) $1 \frac { 1 } { 5 } \mathrm { hr }$
D) $3 \frac { 1 } { 5 } \mathrm { hr }$
E) $1 \frac { 1 } { 3 } \mathrm { hr }$

Question 57

The focal length f of a lens is given by the formula where d1 is the distance from the object to the lens and d2 is the distance from the lens to the image. Solve the formula for f .

Question 58

Solve the equation and check the solution. $$\frac { x } { 7 } - \frac { x } { 5 } = - 6$$

A)- 105
B)105
C)210
D)- 210
E)no solution

Question 59

A plane can fly 260 miles in the same time as it takes a car to go 110 miles. If the car travels 90 mph slower than the plane, find the speed of the plane.

A)116 mph
B)146 mph
C)126 mph
D)156 mph
E)136 mph

Question 60

If the same number is added to both the numerator and the denominator of $\frac { 1 } { 9 }$ , the result is $\frac { 3 } { 11 }$ .Find the number.

A)1
B)4
C)0
D)2
E)3

Question 61

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 = 13, b = 4, c = 34

A) $15$ ft
B) $8$ ft
C) $10 \frac { 6 } { 13 } \mathrm { ft }$
D) $7 \frac { 6 } { 13 } \mathrm { ft }$

Question 62

Express the sentence as a formula. $Z$ varies inversely with the cube of $p$ .

A) $z = k p ^ { 3 }$
B) $z = k / p ^ { 3 }$
C) $z ^ { 3 } = k / p ^ { 3 }$
D) $z ^ { 3 } = k p ^ { 3 }$

Question 63

Solve the proportion for the variable. z = __________

Question 64

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 65

Express the sentence as a formula. Assume that $k$ is a constant of variation. For a fixed area, the length l of a rectangle varies inversely with its width $y$ .

A) $l = k y$
B) $l = \sqrt [ k ] { y }$
C) $l = k + y$
D) $l = y ^ { k }$
E) $l = \frac { k } { y }$

Question 66

Write the ratio 27 feet to 12 yards as a fraction in simplest form. ( Hint : 3 feet = 1 yard.)

A) $\frac { 3 } { 4 }$
B) $\frac { 4 } { 3 }$
C) $4$
D) $\frac { 1 } { 3 }$
E) $3$

Question 67

Assume that all variables represent positive numbers. Assume that $d$ varies directly with t . If $d = 24$ when $t = 4$ , find t when $d = 4$ .

A) $t = \frac { 2 } { 3 }$
B) $t = \frac { 1 } { 3 }$
C) $t = \frac { 3 } { 2 }$
D) $t = 2$
E) $t =$

Question 68

Solve for the variable in the proportion. $$\frac { 3 } { 5 } = \frac { x } { 15 }$$

A) $x = 7$
B) $x = 9$
C) $x = 10$
D) $x = 12$
E) $x = 6$

Question 69

A model railroad engine is 2 inches long. If the scale is 168 feet to 1 foot, how long is the real engine?

A)31
B)29
C)26
D)34
E)28

Question 70

A school board has determined that there should be 3 teachers for every 50 students. How many teachers are needed for an enrollment of 2,750 students?

A)190 teachers
B)176 teachers
C)165 teachers
D)181 teachers
E)222 teachers

Question 71

A recipe for spaghetti sauce requires 6 bottles of ketchup to make 8 gallons of sauce. How many bottles of ketchup are needed to make 56 gallons of sauce?

Question 72

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 73

Assume that all variables represent positive numbers. Assume that $r$ varies directly with $s$ . If $r = 10$ when $s = 4$ , find $s$ when $S = 12$ .

A) $r = 29$
B) $r = 28$
C) $r = 27$
D) $r = 30$
E) $r = 26$

Question 74

Express the sentence as a formula. Assume that $k$ is a constant of variation. The distance $S$ that a body falls varies directly with the square of the time $z$ .

A) $s = \frac { k } { z ^ { 2 } }$
B) $s = k + z ^ { 2 }$
C) $s = k z ^ { 2 }$
D) $s = k - z ^ { 2 }$
E) $s = \frac { z ^ { 2 } } { k }$

Question 75

Solve the proportion for the variable. y = __________