Question 1

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

Accepted Answer

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

Question 2

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.

Accepted Answer

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

Question 3

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

Accepted Answer

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)  $\frac { 3 ( x - 4 ) } { x + 6 }$ 
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 4

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

Accepted Answer

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

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?

Accepted Answer

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)  $1 \frac { 1 } { 3 } \mathrm { hr }$ 
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 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

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)  $\frac { 2 x + 20 } { 13 }$ 
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 8

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.

Accepted Answer

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

Question 9

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

Accepted Answer

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. 
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 10

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

Accepted Answer

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)  $\frac { 18 } { 19 } \cdot \frac { x w } { y z }$ 
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 11

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

Accepted Answer

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)  $\frac { x - 16 } { x ( x + 8 ) }$ 
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 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

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 
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 14

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?

Accepted Answer

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)  $26 \frac { 15 } { 22 } \mathrm {~min}$ 
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 15

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. $\text { Toffee }$

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

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

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

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

Accepted Answer

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

Question 16

Perform the indicated operation. Simplify the answer if possible. $$\frac { 5 ( a + 9 ) } { a - 9 } - \frac { a + 10 } { 9 - a }$$

Accepted Answer

The answer of Perform the indicated operation. Simplify the answer...

Question 17

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

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

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

Question 20

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

Accepted Answer

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)  $\frac { z + 6 } { v + 6 }$ 
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)  $\frac { z + 6 } { v + 6 }$

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

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.

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

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

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?

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

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

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.

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

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

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

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

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

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?

Perform the indicated operation. Simplify the answer if possible. $\frac { 5 ( a + 9 ) } { a - 9 } - \frac { a + 10 } { 9 - a }$

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

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

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

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

An Introduction to Algebra

Equations, Inequalities, and Problem Solving

Graphs, Linear Equations, and Inequalities in Two Variables; Functions From Campus to Careers: Physical Therapis

Solving Systems of Equations and Inequalities

Exponents and Polynomials

Factoring and Quadratic Equations

Roots and Radicals

Quadratic Equations

Filters

Exam 7: Rational Expressions and Equations

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

If a number is added to the numerator of 38\frac { 3 } { 8 }83​ and twice as much is added to the denominator, the result is 49\frac { 4 } { 9 }94​ . Find the number.

Add the rational expressions. Be sure to express your answer in simplest form. 3xx−6+12x−6\frac { 3 x } { x - 6 } + \frac { 12 } { x - 6 }x−63x​+x−612​

Perform the indicated operation and simplify. p5r3cd⋅c−4d5p\frac { p ^ { 5 } r ^ { 3 } } { c d } \cdot \frac { c ^ { - 4 } d ^ { 5 } } { p }cdp5r3​⋅pc−4d5​

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?

Solve the equation and check the solution. x7−x5=−6\frac { x } { 7 } - \frac { x } { 5 } = - 67x​−5x​=−6

Add the rational expressions. Be sure to express your answer in simplest form. x−46+x+87\frac { x - 4 } { 6 } + \frac { x + 8 } { 7 }6x−4​+7x+8​

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.

Simplify this rational expression. Assume the denominator never equals zero. 3−3⋅yy2−1\frac { 3 - 3 \cdot y } { y ^ { 2 } - 1 }y2−13−3⋅y​

Solve the expression. 18xy⋅z19w=??(y≠0,w≠0)\frac { 18 x } { y } \cdot \frac { z } { 19 w } = \frac { ? } { ? } ( y \neq 0 , w \neq 0 )y18x​⋅19wz​=??​(y=0,w=0)

Perform the indicated operation and express your answer in simplest form. 3xx2+8x−2x\frac { 3 x } { x ^ { 2 } + 8 x } - \frac { 2 } { x }x2+8x3x​−x2​

Simplify this rational expression. Assume the denominator never equals zero. 8x−2a2a−8x\frac { 8 x - 2 a } { 2 a - 8 x }2a−8x8x−2a​

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

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?

Perform the indicated operation. Simplify the answer if possible. 5(a+9)a−9−a+109−a\frac { 5 ( a + 9 ) } { a - 9 } - \frac { a + 10 } { 9 - a }a−95(a+9)​−9−aa+10​

Perform the division. (3y−12+15y2)÷(4−5y)\left( 3 y - 12 + 15 y ^ { 2 } \right) \div ( 4 - 5 y )(3y−12+15y2)÷(4−5y)

Express the sentence as a formula. ZZZ varies inversely with the cube of ppp .

Solve the equation and check the solution. 3a4+a2=−20\frac { 3 a } { 4 } + \frac { a } { 2 } = - 2043a​+2a​=−20

Perform the indicated operation and simplify. z+16v2+22v+96⋅(v+6)\frac { z + 16 } { v ^ { 2 } + 22 v + 96 } \cdot ( v + 6 )v2+22v+96z+16​⋅(v+6)

An Introduction to Algebra

Equations, Inequalities, and Problem Solving

Graphs, Linear Equations, and Inequalities in Two Variables; Functions From Campus to Careers: Physical Therapis

Solving Systems of Equations and Inequalities

Exponents and Polynomials

Factoring and Quadratic Equations

Roots and Radicals

Quadratic Equations

Filters

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

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.

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

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

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

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

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

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

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

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

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

Perform the indicated operation. Simplify the answer if possible. $\frac { 5 ( a + 9 ) } { a - 9 } - \frac { a + 10 } { 9 - a }$

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

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

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

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