Solve Equations with Fractions

Determine Whether a Fraction is a Solution of an Equation

As we saw in Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality, a solution of an equation is a value that makes a true statement when substituted for the variable in the equation. In those sections, we found whole number and integer solutions to equations. Now that we have worked with fractions, we are ready to find fraction solutions to equations.

The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, or a fraction.

Determine whether each of the following is a solution of $x - \frac{3}{10} = \frac{1}{2} .$

ⓐ $x = 1$
ⓑ $x = \frac{4}{5}$
ⓒ $x = - \frac{4}{5}$

Solution

ⓐ


Change to fractions with a LCD of 10.
Subtract.

Since $x = 1$ does not result in a true equation, $1$ is not a solution to the equation.

ⓑ
	$The image displays the algebraic equation x - 3/10 = 1/2, where 'x' is an unknown variable and the fractions 3/10 and 1/2 are constants. This equation requires solving for the value of x.$
$The image shows the text 'Substitute 4/5 for x.' in a mathematical context, likely instructing to replace the variable 'x' with the fraction '4/5'.$	$A mathematical equation asking whether 4/5 minus 3/10 equals 1/2. The fraction 4/5 is colored red.$
	$The image displays a fractional subtraction problem: 8/10 - 3/10. A question mark over the equals sign asks if the result is 5/10, evaluating the truth of the statement.$
Subtract.

Since $x = \frac{4}{5}$ results in a true equation, $\frac{4}{5}$ is a solution to the equation $x - \frac{3}{10} = \frac{1}{2} .$

ⓒ

$The image shows the instruction 'Substitute -4/5 for x.' with the fraction -4/5 written in red.$	$A mathematical expression shows the equation: negative four-fifths minus three-tenths equals, with a question mark over the equal sign, one-half. The fraction negative four-fifths is in red.$
	$A mathematical equation shows '-8/10 - 3/10' on the left, an equals sign with a question mark above it in the middle, and '5/10' on the right. The first fraction's numerator and denominator are red.$
Subtract.

Since $x = - \frac{4}{5}$ does not result in a true equation, $- \frac{4}{5}$ is not a solution to the equation.

Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

In Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality, we solved equations using the Addition, Subtraction, and Division Properties of Equality. We will use these same properties to solve equations with fractions.

Addition, Subtraction, and Division Properties of Equality

For any numbers $a, b,$ and $c,$

This table lists fundamental properties of equality including addition, subtraction, and division, showing their mathematical statements.
if $a = b,$ then $a + c = b + c .$	Addition Property of Equality
if $a = b,$ then $a - c = b - c .$	Subtraction Property of Equality
if $a = b,$ then $\frac{a}{c} = \frac{b}{c}, c \neq 0 .$	Division Property of Equality

In other words, when you add or subtract the same quantity from both sides of an equation, or divide both sides by the same quantity, you still have equality.

Solve: $y + \frac{9}{16} = \frac{5}{16} .$

Solution


Subtract $\frac{9}{16}$ from each side to undo the addition.
Simplify on each side of the equation.
Simplify the fraction.
Check:
Substitute $y = - \frac{1}{4}$ .	$A mathematical equation shows -1/4 + 9/16 =? 5/16, representing the sum of two fractions with a question mark indicating a verification or unknown result.$
Rewrite as fractions with the LCD.	$The mathematical expression -4/16 + 9/16 ?= 5/16, asking to verify the equality of the fractional sum.$
Add.	$The image shows the fraction 5/16 equals 5/16, followed by a checkmark, indicating correctness or verification of the equality.$

Since $y = - \frac{1}{4}$ makes $y + \frac{9}{16} = \frac{5}{16}$ a true statement, we know we have found the solution to this equation.

We used the Subtraction Property of Equality in Example 2. Now we’ll use the Addition Property of Equality.

Solve: $a - \frac{5}{9} = - \frac{8}{9} .$

Solution

		$A mathematical equation is displayed on a white background. The equation reads as 'a - 5/9 = -8/9', where 'a' is a variable, and '5/9' and '-8/9' are fractions.$
Add $\frac{5}{9}$ from each side to undo the subtraction.		$The image shows a mathematical equation: a - 5/9 + 5/9 = -8/9 + 5/9. The fractions 5/9 on both sides of the equation are highlighted in red.$
Simplify on each side of the equation.
Simplify the fraction.
Check:
Substitute $a = - \frac{1}{3}$ .	$A mathematical equation with fractions: -1/3 (in red) - 5/9 with a question mark above the equals sign, followed by -8/9. It asks to verify the equality of the expression.$
Change to common denominator.	$The equation -3/9 - 5/9 ?= -8/9, where a question mark above the equals sign prompts verification of the fractional subtraction.$
Subtract.

Since $a = - \frac{1}{3}$ makes the equation true, we know that $a = - \frac{1}{3}$ is the solution to the equation.

The next example may not seem to have a fraction, but let’s see what happens when we solve it.

Solve: $10 q = 44 .$

Solution

Illustrates the step-by-step process of solving the linear equation 10q=44 and verifying its solution.
		$10 q = 44$
Divide both sides by 10 to undo the multiplication.		$\frac{10 q}{10} = \frac{44}{10}$
Simplify.		$q = \frac{22}{5}$
Check:
Substitute $q = \frac{22}{5}$ into the original equation.	$10 (\frac{22}{5}) \overset{?}{=} 44$
Simplify.	$\overset{2}{10} (\frac{22}{5}) \overset{?}{=} 44$
Multiply.	$44 = 44 ✓$

The solution to the equation was the fraction $\frac{22}{5} .$ We leave it as an improper fraction.

Solve Equations with Fractions Using the Multiplication Property of Equality

Consider the equation $\frac{x}{4} = 3 .$ We want to know what number divided by $4$ gives $3 .$ So to “undo” the division, we will need to multiply by $4 .$ The Multiplication Property of Equality will allow us to do this. This property says that if we start with two equal quantities and multiply both by the same number, the results are equal.

Let’s use the Multiplication Property of Equality to solve the equation $\frac{x}{7} = −9 .$

Solve: $\frac{x}{7} = −9 .$

Solution


Use the Multiplication Property of Equality to multiply both sides by $7$ . This will isolate the variable.
Multiply.
Simplify.
	$A mathematical equation shows the fraction negative sixty-three over seven, followed by a question mark above an equals sign, and then negative nine. This setup asks if negative sixty-three divided by seven is equal to negative nine.$
The equation is true.

Solve: $\frac{p}{−8} = −40 .$

Solution

Here, $p$ is divided by $−8 .$ We must multiply by $−8$ to isolate $p .$


Multiply both sides by $−8$
Multiply.		$A mathematical equation shows a fraction with '-8p' in the numerator and '-8' in the denominator, set equal to '320'.$
Simplify.
Check:
Substitute $p = 320$ .
The equation is true.

Solve Equations with a Coefficient of $−1$

Look at the equation $- y = 15 .$ Does it look as if $y$ is already isolated? But there is a negative sign in front of $y,$ so it is not isolated.

There are three different ways to isolate the variable in this type of equation. We will show all three ways in Example 7.

Solve: $- y = 15 .$

Solution

One way to solve the equation is to rewrite $- y$ as $−1 y,$ and then use the Division Property of Equality to isolate $y .$


Rewrite $- y$ as $−1 y$ .
Divide both sides by −1.
Simplify each side.

Another way to solve this equation is to multiply both sides of the equation by $−1 .$


Multiply both sides by −1.
Simplify each side.

The third way to solve the equation is to read $- y$ as “the opposite of $y$ .” What number has $15$ as its opposite? The opposite of $15$ is $−15 .$ So $y = −15 .$

For all three methods, we isolated $y$ is isolated and solved the equation.

Check:


Substitute $y = −15$ .
Simplify. The equation is true.

Solve Equations with a Fraction Coefficient

When we have an equation with a fraction coefficient we can use the Multiplication Property of Equality to make the coefficient equal to $1 .$

For example, in the equation:

\frac{3}{4} x = 24

The coefficient of $x$ is $\frac{3}{4} .$ To solve for $x,$ we need its coefficient to be $1 .$ Since the product of a number and its reciprocal is $1,$ our strategy here will be to isolate $x$ by multiplying by the reciprocal of $\frac{3}{4} .$ We will do this in Example 8.

Solve: $\frac{3}{4} x = 24 .$

Solution

		$A mathematical equation is displayed on a white background: 3/4x = 24. The equation represents a linear algebraic problem involving a fraction and a variable.$
Multiply both sides by the reciprocal of the coefficient.		$An algebraic equation is shown: (4/3) multiplied by (3/4)x equals (4/3) multiplied by 24. The fraction 4/3 appears in red on both sides of the equation.$
Simplify.		$A mathematical equation showing 1x equals the product of 4/3 and 24/1, which simplifies to 1x = 32. It demonstrates a step in solving for 'x' by multiplying fractions.$
Multiply.
Check:
Substitute $x = 32$ .
Rewrite $32$ as a fraction.
Multiply. The equation is true.

Notice that in the equation $\frac{3}{4} x = 24,$ we could have divided both sides by $\frac{3}{4}$ to get $x$ by itself. Dividing is the same as multiplying by the reciprocal, so we would get the same result. But most people agree that multiplying by the reciprocal is easier.

Solve: $- \frac{3}{8} w = 72 .$

Solution

The coefficient is a negative fraction. Remember that a number and its reciprocal have the same sign, so the reciprocal of the coefficient must also be negative.


Multiply both sides by the reciprocal of $− \frac{3}{8}$ .		$The equation -8/3(-3/8w) = (-8/3)72 is displayed, illustrating a linear equation involving fractions and the variable 'w'.$
Simplify; reciprocals multiply to one.		$The equation shows '1w = -8/3 ×72/1', representing a multiplication of two fractions, one negative and one positive, to solve for 'w'.$
Multiply.
Check:
Let $w = −192$ .
Multiply. It checks.

Translate Sentences to Equations and Solve

Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference.

Subtraction Property of Equality: For any real numbers $a, b,$ and $c,$ if $a = b,$ then $a - c = b - c .$	Addition Property of Equality: For any real numbers $a, b,$ and $c,$ if $a = b,$ then $a + c = b + c .$
Division Property of Equality: For any numbers $a, b,$ and $c,$ where $c \neq 0$ if $a = b,$ then $\frac{a}{c} = \frac{b}{c}$	Multiplication Property of Equality: For any real numbers $a, b,$ and $c$ if $a = b,$ then $a c = b c$

When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.

In the next few examples, we’ll translate sentences into equations and then solve the equations. It might be helpful to review the translation table in Evaluate, Simplify, and Translate Expressions.

Translate and solve: $n$ divided by $6$ is $−24 .$

Solution

Translate.
Multiply both sides by $6$ .
Simplify.
Check:	Is $−144$ divided by $6$ equal to $−24$ ?
Translate.	$A math problem displays the fraction -144/6, followed by a question mark above an equals sign, and then -24. It asks whether -144 divided by 6 is equal to -24.$
Simplify. It checks.

Translate and solve: The quotient of $q$ and $−5$ is $70 .$

Solution

Translate.
Multiply both sides by $−5$ .
Simplify.
Check:	Is the quotient of $−350$ and $−5$ equal to $70$ ?
Translate.
Simplify. It checks.

Translate and solve: Two-thirds of $f$ is $18 .$

Solution

Translate.
Multiply both sides by $\frac{3}{2}$ .		$A math equation shows (3/2) * (2/3)f = (3/2) * 18, with the red fraction 3/2 applied to both sides, likely to solve for 'f' after simplification.$
Simplify.
Check:	Is two-thirds of $27$ equal to $18$ ?
Translate.
Simplify. It checks.

Translate and solve: The quotient of $m$ and $\frac{5}{6}$ is $\frac{3}{4} .$

Solution

This table demonstrates the step-by-step process of solving and verifying an algebraic equation involving fractions for the variable 'm'.
		The quotient of $m$ and $\frac{5}{6}$ is $\frac{3}{4}$ .
Translate.		$\frac{m}{\frac{5}{6}} = \frac{3}{4}$
Multiply both sides by $\frac{5}{6}$ to isolate $m$ .		$\frac{5}{6} (\frac{m}{\frac{5}{6}}) = \frac{5}{6} (\frac{3}{4})$
Simplify.		$m = \frac{5 \cdot 3}{6 \cdot 4}$
Remove common factors and multiply.		$m = \frac{5}{8}$
Check:
Is the quotient of $\frac{5}{8}$ and $\frac{5}{6}$ equal to $\frac{3}{4}$ ?	$\frac{\frac{5}{8}}{\frac{5}{6}} \overset{?}{=} \frac{3}{4}$
Rewrite as division.	$\frac{5}{8} \div \frac{5}{6} \overset{?}{=} \frac{3}{4}$
Multiply the first fraction by the reciprocal of the second.	$\frac{5}{8} \cdot \frac{6}{5} \overset{?}{=} \frac{3}{4}$
Simplify.	$\frac{3}{4} = \frac{3}{4} ✓$

Our solution checks.

Translate and solve: The sum of three-eighths and $x$ is three and one-half.

Solution

Translate.
Use the Subtraction Property of Equality to subtract $\frac{3}{8}$ from both sides.
Combine like terms on the left side.
Convert mixed number to improper fraction.
Convert to equivalent fractions with LCD of 8.	$A mathematical equation shows x equals 28 over 8 minus 3 over 8, demonstrating subtraction of fractions with a common denominator.$
Subtract.
Write as a mixed number.

We write the answer as a mixed number because the original problem used a mixed number.

Check:

Is the sum of three-eighths and $3 \frac{1}{8}$ equal to three and one-half?

Step-by-step verification of a fractional addition problem.
	$\frac{3}{8} + 3 \frac{1}{8} \overset{?}{=} 3 \frac{1}{2}$
Add.	$3 \frac{4}{8} \overset{?}{=} 3 \frac{1}{2}$
Simplify.	$3 \frac{1}{2} = 3 \frac{1}{2} ✓$

The solution checks.

Key Concepts

Determine whether a number is a solution to an equation.
1. Substitute the number for the variable in the equation.
2. Simplify the expressions on both sides of the equation.
3. Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.
Addition, Subtraction, and Division Properties of Equality
- For any numbers a, b, and c,
  if $a = b$ , then $a + c = b + c$ . Addition Property of Equality
- if $a = b$ , then $a - c = b - c$ . Subtraction Property of Equality
- if $a = b$ , then $\frac{a}{c} = \frac{b}{c}$ , $c \neq 0$ . Division Property of Equality
The Multiplication Property of Equality
- For any numbers $a b$ and $c, a = b$ , then $a c = b c$ .
- If you multiply both sides of an equation by the same quantity, you still have equality.

Section Exercises

Practice Makes Perfect

Determine Whether a Fraction is a Solution of an Equation

In the following exercises, determine whether each number is a solution of the given equation.

x - \frac{2}{5} = \frac{1}{10}

ⓐ $x = 1$
ⓑ $x = \frac{1}{2}$
ⓒ $x = - \frac{1}{2}$

y - \frac{1}{3} = \frac{5}{12}

ⓐ $y = 1$
ⓑ $y = \frac{3}{4}$
ⓒ $y = - \frac{3}{4}$

Solution

ⓐ no
ⓑ yes
ⓒ no

h + \frac{3}{4} = \frac{2}{5}

ⓐ $h = 1$
ⓑ $h = \frac{7}{20}$
ⓒ $h = - \frac{7}{20}$

k + \frac{2}{5} = \frac{5}{6}

ⓐ $k = 1$
ⓑ $k = \frac{13}{30}$
ⓒ $k = - \frac{13}{30}$

Solution

ⓐ no
ⓑ yes
ⓒ no

Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

In the following exercises, solve.

$y + \frac{1}{3} = \frac{4}{3}$

$m + \frac{3}{8} = \frac{7}{8}$

Solution

$m = \frac{1}{2}$

$f + \frac{9}{10} = \frac{2}{5}$

$h + \frac{5}{6} = \frac{1}{6}$

Solution

$h = - \frac{2}{3}$

$a - \frac{5}{8} = - \frac{7}{8}$

$c - \frac{1}{4} = - \frac{5}{4}$

Solution

c = −1

$x - (- \frac{3}{20}) = - \frac{11}{20}$

$z - (- \frac{5}{12}) = - \frac{7}{12}$

Solution

z = −1

$n - \frac{1}{6} = \frac{3}{4}$

$p - \frac{3}{10} = \frac{5}{8}$

Solution

$p = \frac{37}{40}$

$s + (- \frac{1}{2}) = - \frac{8}{9}$

$k + (- \frac{1}{3}) = - \frac{4}{5}$

Solution

$k = - \frac{7}{15}$

$5 j = 17$

$7 k = 18$

Solution

$k = \frac{18}{7}$

$−4 w = 26$

$−9 v = 33$

Solution

$v = - \frac{11}{3}$

Solve Equations with Fractions Using the Multiplication Property of Equality

In the following exercises, solve.

$\frac{f}{4} = −20$

$\frac{b}{3} = −9$

Solution

b = −27

$\frac{y}{7} = −21$

$\frac{x}{8} = −32$

Solution

x = −256

$\frac{p}{−5} = −40$

$\frac{q}{−4} = −40$

Solution

q = 160

$\frac{r}{−12} = −6$

$\frac{s}{−15} = −3$

Solution

s = 45

$- x = 23$

$- y = 42$

Solution

y = −42

$- h = - \frac{5}{12}$

$- k = - \frac{17}{20}$

Solution

$k = \frac{17}{20}$

$\frac{4}{5} n = 20$

$\frac{3}{10} p = 30$

Solution

p = 100

$\frac{3}{8} q = −48$

$\frac{5}{2} m = −40$

Solution

m = −16

$- \frac{2}{9} a = 16$

$- \frac{3}{7} b = 9$

Solution

b = −21

$- \frac{6}{11} u = −24$

$- \frac{5}{12} v = −15$

Solution

v = 36

Mixed Practice

In the following exercises, solve.

$3 x = 0$

$8 y = 0$

Solution

y = 0

$4 f = \frac{4}{5}$

$7 g = \frac{7}{9}$

Solution

$g = \frac{1}{9}$

$p + \frac{2}{3} = \frac{1}{12}$

$q + \frac{5}{6} = \frac{1}{12}$

Solution

$q = - \frac{3}{4}$

$\frac{7}{8} m = \frac{1}{10}$

$\frac{1}{4} n = \frac{7}{10}$

Solution

$n = \frac{14}{5}$

$- \frac{2}{5} = x + \frac{3}{4}$

$- \frac{2}{3} = y + \frac{3}{8}$

Solution

$y = - \frac{25}{24}$

$\frac{11}{20} = - f$

$\frac{8}{15} = - d$

Solution

$d = - \frac{8}{15}$

Translate Sentences to Equations and Solve

In the following exercises, translate to an algebraic equation and solve.

$n$ divided by eight is $−16 .$

$n$ divided by six is $−24 .$

Solution

$\frac{n}{6} = −24; n = −144$

$m$ divided by $−9$ is $−7 .$

$m$ divided by $−7$ is $−8 .$

Solution

$\frac{m}{−7} = −8; m = 56$

The quotient of $f$ and $−3$ is $−18 .$

The quotient of $f$ and $−4$ is $−20 .$

Solution

$\frac{f}{−4} = −20; f = 80$

The quotient of $g$ and twelve is $8 .$

The quotient of $g$ and nine is $14 .$

Solution

$\frac{g}{9} = 14; g = 126$

Three-fourths of $q$ is $12 .$

Two-fifths of $q$ is $20 .$

Solution

$\frac{2}{5} q = 20; q = 50$

Seven-tenths of $p$ is $−63 .$

Four-ninths of $p$ is $−28 .$

Solution

$\frac{4}{9} p = −28; p = −63$

$m$ divided by $4$ equals negative $6 .$

The quotient of $h$ and $2$ is $43 .$

Solution

$\frac{h}{2} = 43; h = 86$

Three-fourths of $z$ is $15 .$

The quotient of $a$ and $\frac{2}{3}$ is $\frac{3}{4} .$

Solution

$\frac{a}{\frac{2}{3}} = \frac{3}{4}; a = \frac{1}{2}$

The sum of five-sixths and $x$ is $\frac{1}{2} .$

The sum of three-fourths and $x$ is $\frac{1}{8} .$

Solution

$\frac{3}{4} + x = \frac{1}{8}; x = - \frac{5}{8}$

The difference of $y$ and one-fourth is $- \frac{1}{8} .$

The difference of $y$ and one-third is $- \frac{1}{6} .$

Solution

$y - \frac{1}{3} = - \frac{1}{6}; y = \frac{1}{6}$

Everyday Math

Shopping Teresa bought a pair of shoes on sale for $$ 48$ . The sale price was $\frac{2}{3}$ of the regular price. Find the regular price of the shoes by solving the equation $\frac{2}{3} p = 48$

Playhouse The table in a child’s playhouse is $\frac{3}{5}$ of an adult-size table. The playhouse table is $18$ inches high. Find the height of an adult-size table by solving the equation $\frac{3}{5} h = 18 .$

Solution

30 inches

Writing Exercises

Example 6 describes three methods to solve the equation $- y = 15 .$ Which method do you prefer? Why?

Richard thinks the solution to the equation $\frac{3}{4} x = 24$ is $16 .$ Explain why Richard is wrong.

Solution

Answers will vary.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

A self-assessment chart with 'I can...' statements related to solving equations with fractions, including determining solutions, using properties of equality, and translating sentences, with columns for confidence levels: Confidently, With some help, and No-I don't get it!

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

Chapter Review Exercises

Visualize Fractions

In the following exercises, name the fraction of each figure that is shaded.

A circle is shown. It is divided into 8 equal pieces. 5 pieces are shaded.

A square is shown. It is divided into 9 equal pieces. 5 pieces are shaded.

Solution

$\frac{5}{9}$

In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number.

Two squares are shown. Both are divided into four equal pieces. The square on the left has all 4 pieces shaded. The square on the right has one piece shaded.

Two circles are shown. Both are divided into two equal pieces. The circle on the left has both pieces shaded. The circle on the right has one piece shaded.

Solution

$\frac{3}{2} = 1 \frac{1}{2}$

In the following exercises, convert the improper fraction to a mixed number.

$\frac{58}{15}$

$\frac{63}{11}$

Solution

$5 \frac{8}{11}$

In the following exercises, convert the mixed number to an improper fraction.

$12 \frac{1}{4}$

$9 \frac{4}{5}$

Solution

$\frac{49}{5}$

Find three fractions equivalent to $\frac{2}{5} .$ Show your work, using figures or algebra.

Find three fractions equivalent to $- \frac{4}{3} .$ Show your work, using figures or algebra.

Solution

Answers may vary.

In the following exercises, locate the numbers on a number line.

$\frac{5}{8}, \frac{4}{3}, 3 \frac{3}{4}, 4$

$\frac{1}{4}, - \frac{1}{4}, 1 \frac{1}{3}, −1 \frac{1}{3}, \frac{7}{2}, - \frac{7}{2}$

Solution

In the following exercises, order each pair of numbers, using $<$ or $>.$

$−1___- \frac{2}{5}$

$−2 \frac{1}{2}___−3$

Solution

Multiply and Divide Fractions

In the following exercises, simplify.

$- \frac{63}{84}$

$- \frac{90}{120}$

Solution

$- \frac{3}{4}$

$- \frac{14 a}{14 b}$

$- \frac{8 x}{8 y}$

Solution

$- \frac{x}{y}$

In the following exercises, multiply.

$\frac{2}{5} \cdot \frac{8}{13}$

$- \frac{1}{3} \cdot \frac{12}{7}$

Solution

$- \frac{4}{7}$

$\frac{2}{9} \cdot (- \frac{45}{32})$

$6 m \cdot \frac{4}{11}$

Solution

$\frac{24 m}{11}$

$- \frac{1}{4} (−32)$

$\frac{16}{5} \cdot \frac{15}{8}$

Solution

In the following exercises, find the reciprocal.

$\frac{2}{9}$

$\frac{15}{4}$

Solution

$\frac{4}{15}$

$3$

$- \frac{1}{4}$

Solution

−4

Fill in the chart.

	Opposite	Absolute Value	Reciprocal
$- \frac{5}{13}$
$\frac{3}{10}$
$\frac{9}{4}$
$−12$

In the following exercises, divide.

$\frac{2}{3} \div \frac{1}{6}$

Solution

$(- \frac{3 x}{5}) \div (- \frac{2 y}{3})$

$\frac{4}{5} \div 3$

Solution

$\frac{4}{15}$

$8 \div \frac{8}{3}$

$\frac{5}{18} \div (- \frac{b}{9})$

Solution

$- \frac{5}{2 b}$

Multiply and Divide Mixed Numbers and Complex Fractions

In the following exercises, perform the indicated operation.

$3 \frac{1}{5} \cdot 1 \frac{7}{8}$

$−5 \frac{7}{12} \cdot 4 \frac{4}{11}$

Solution

$- \frac{268}{11}$

$8 \div 2 \frac{2}{3}$

$8 \frac{2}{3} \div 1 \frac{1}{12}$

Solution

In the following exercises, translate the English phrase into an algebraic expression.

the quotient of $8$ and $y$

the quotient of $V$ and the difference of $h$ and $6$

Solution

$\frac{V}{h - 6}$

In the following exercises, simplify the complex fraction

$\frac{\frac{5}{8}}{\frac{4}{5}}$

$\frac{\frac{8}{9}}{−4}$

Solution

$- \frac{2}{9}$

$\frac{\frac{n}{4}}{\frac{3}{8}}$

$\frac{−1 \frac{5}{6}}{- \frac{1}{12}}$

Solution

In the following exercises, simplify.

$\frac{5 + 16}{5}$

$\frac{8 \cdot 4 - 5^{2}}{3 \cdot 12}$

Solution

$\frac{7}{36}$

$\frac{8 \cdot 7 + 5 (8 - 10)}{9 \cdot 3 - 6 \cdot 4}$

Add and Subtract Fractions with Common Denominators

In the following exercises, add.

$\frac{3}{8} + \frac{2}{8}$

Solution

$\frac{5}{8}$

$\frac{4}{5} + \frac{1}{5}$

$\frac{2}{5} + \frac{1}{5}$

Solution

$\frac{3}{5}$

$\frac{15}{32} + \frac{9}{32}$

$\frac{x}{10} + \frac{7}{10}$

Solution

$\frac{x + 7}{10}$

In the following exercises, subtract.

$\frac{8}{11} - \frac{6}{11}$

$\frac{11}{12} - \frac{5}{12}$

Solution

$\frac{1}{2}$

$\frac{4}{5} - \frac{y}{5}$

$- \frac{31}{30} - \frac{7}{30}$

Solution

$- \frac{19}{15}$

$\frac{3}{2} - (\frac{3}{2})$

$\frac{11}{15} - \frac{5}{15} - (- \frac{2}{15})$

Solution

$\frac{8}{15}$

Add and Subtract Fractions with Different Denominators

In the following exercises, find the least common denominator.

$\frac{1}{3}$ and $\frac{1}{12}$

$\frac{1}{3}$ and $\frac{4}{5}$

Solution

$\frac{8}{15}$ and $\frac{11}{20}$

$\frac{3}{4}, \frac{1}{6},$ and $\frac{5}{10}$

Solution

In the following exercises, change to equivalent fractions using the given LCD.

$\frac{1}{3}$ and $\frac{1}{5},$ LCD $= 15$

$\frac{3}{8}$ and $\frac{5}{6},$ LCD $= 24$

Solution

$\frac{9}{24}$ and $\frac{20}{24}$

$- \frac{9}{16}$ and $\frac{5}{12},$ LCD $= 48$

$\frac{1}{3}, \frac{3}{4}$ and $\frac{4}{5},$ LCD $= 60$

Solution

$\frac{20}{60}, \frac{45}{60}$ and $\frac{48}{60}$

In the following exercises, perform the indicated operations and simplify.

$\frac{1}{5} + \frac{2}{3}$

$\frac{11}{12} - \frac{2}{3}$

Solution

$\frac{1}{4}$

$- \frac{9}{10} - \frac{3}{4}$

$- \frac{11}{36} - \frac{11}{20}$

Solution

$- \frac{77}{90}$

$- \frac{22}{25} + \frac{9}{40}$

$\frac{y}{10} - \frac{1}{3}$

Solution

$\frac{3 y - 10}{30}$

$\frac{2}{5} + (- \frac{5}{9})$

$\frac{4}{11} \div \frac{2}{7 d}$

Solution

$\frac{14 d}{11}$

$\frac{2}{5} + (- \frac{3 n}{8}) (- \frac{2}{9 n})$

$\frac{{(\frac{2}{3})}^{2}}{{(\frac{5}{8})}^{2}}$

Solution

$\frac{256}{225}$

$(\frac{11}{12} + \frac{3}{8}) \div (\frac{5}{6} - \frac{1}{10})$

In the following exercises, evaluate.

y - \frac{4}{5}

when

ⓐ $y = - \frac{4}{5}$
ⓑ $y = \frac{1}{4}$

Solution

ⓐ $- \frac{8}{5}$
ⓑ $- \frac{11}{20}$

$6 m n^{2}$ when $m = \frac{3}{4}$ and $n = - \frac{1}{3}$

Add and Subtract Mixed Numbers

In the following exercises, perform the indicated operation.

$4 \frac{1}{3} + 9 \frac{1}{3}$

Solution

$13 \frac{2}{3}$

$6 \frac{2}{5} + 7 \frac{3}{5}$

$5 \frac{8}{11} + 2 \frac{4}{11}$

Solution

$8 \frac{1}{11}$

$3 \frac{5}{8} + 3 \frac{7}{8}$

$9 \frac{13}{20} - 4 \frac{11}{20}$

Solution

$5 \frac{1}{10}$

$2 \frac{3}{10} - 1 \frac{9}{10}$

$2 \frac{11}{12} - 1 \frac{7}{12}$

Solution

$1 \frac{1}{3}$

$8 \frac{6}{11} - 2 \frac{9}{11}$

Solve Equations with Fractions

In the following exercises, determine whether the each number is a solution of the given equation.

$x - \frac{1}{2} = \frac{1}{6}$ :

ⓐ $x = 1$
ⓑ $x = \frac{2}{3}$

Solution

ⓐ no
ⓑ yes
ⓒ no

$y + \frac{3}{5} = \frac{5}{9}$ :

ⓐ $y = \frac{1}{2}$
ⓑ $y = \frac{52}{45}$

In the following exercises, solve the equation.

$n + \frac{9}{11} = \frac{4}{11}$

Solution

$n = - \frac{5}{11}$

$x - \frac{1}{6} = \frac{7}{6}$

$h - (- \frac{7}{8}) = - \frac{2}{5}$

Solution

$h = - \frac{51}{40}$

$\frac{x}{5} = −10$

$- z = 23$

Solution

z = −23

In the following exercises, translate and solve.

The sum of two-thirds and $n$ is $- \frac{3}{5} .$

The difference of $q$ and one-tenth is $\frac{1}{2} .$

Solution

$q - \frac{1}{10} = \frac{1}{2}; q = \frac{3}{5}$

The quotient of $p$ and $−4$ is $−8 .$

Three-eighths of $y$ is $24 .$

Solution

$\frac{3}{8} y = 24; y = 64$

Chapter Practice Test

Convert the improper fraction to a mixed number.

$\frac{19}{5}$

Convert the mixed number to an improper fraction.

$3 \frac{2}{7}$

Solution

$\frac{23}{7}$

Locate the numbers on a number line.

$\frac{1}{2}, 1 \frac{2}{3}, −2 \frac{3}{4},$ and $\frac{9}{4}$

In the following exercises, simplify.

$\frac{5}{20}$

Solution

$\frac{1}{4}$

$\frac{18 r}{27 s}$

$\frac{1}{3} \cdot \frac{3}{4}$

Solution

$\frac{1}{4}$

$\frac{3}{5} \cdot 15$

$−36 u (- \frac{4}{9})$

Solution

16u

$−5 \frac{7}{12} \cdot 4 \frac{4}{11}$

$- \frac{5}{6} \div \frac{5}{12}$

Solution

−2

$\frac{7}{11} \div (- \frac{7}{11})$

$\frac{9 a}{10} \div \frac{15 a}{8}$

Solution

$\frac{12}{25}$

$−6 \frac{2}{5} \div 4$

$(−15 \frac{5}{6}) \div (−3 \frac{1}{6})$

Solution

$\frac{−6}{\frac{6}{11}}$

$\frac{\frac{p}{2}}{\frac{q}{5}}$

Solution

$\frac{5 p}{2 q}$

$\frac{- \frac{4}{15}}{−2 \frac{2}{3}}$

$\frac{9^{2} - 4^{2}}{9 - 4}$

Solution

$\frac{2}{d} + \frac{9}{d}$

$- \frac{3}{13} + (- \frac{4}{13})$

Solution

$- \frac{7}{13}$

$- \frac{22}{25} + \frac{9}{40}$

$\frac{2}{5} + (- \frac{7}{5})$

Solution

−1

$- \frac{3}{10} + (- \frac{5}{8})$

$- \frac{3}{4} \div \frac{x}{3}$

Solution

$- \frac{9}{4 x}$

$\frac{2^{3} - 2^{2}}{{(\frac{3}{4})}^{2}}$

$\frac{\frac{5}{14} + \frac{1}{8}}{\frac{9}{56}}$

Solution

Evaluate.

x + \frac{1}{3}

when

ⓐ $x = \frac{2}{3}$
ⓑ $x = - \frac{5}{6}$

In the following exercises, solve the equation.

$y + \frac{3}{5} = \frac{7}{5}$

Solution

$y = \frac{4}{5}$

$a - \frac{3}{10} = - \frac{9}{10}$

$f + (- \frac{2}{3}) = \frac{5}{12}$

Solution

$f = \frac{13}{12}$

$\frac{m}{−2} = −16$

$- \frac{2}{3} c = 18$

Solution

c = −27

Translate and solve: The quotient of $p$ and $−4$ is $−8 .$ Solve for $p .$

Solve Equations with Fractions

Determine Whether a Fraction is a Solution of an Equation

Solution

Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

Solution

Solution

Solution

Solve Equations with Fractions Using the Multiplication Property of Equality

Solution

Solution

Solve Equations with a Coefficient of −1

Solution

Solve Equations with a Fraction Coefficient

Solution

Solution

Translate Sentences to Equations and Solve

Solution

Solution

Solution

Solution

Solution

Key Concepts

Section Exercises

Practice Makes Perfect

Everyday Math

Writing Exercises

Self Check

Chapter Review Exercises

Visualize Fractions

Multiply and Divide Fractions

Multiply and Divide Mixed Numbers and Complex Fractions

Add and Subtract Fractions with Common Denominators

Add and Subtract Fractions with Different Denominators

Add and Subtract Mixed Numbers

Solve Equations with Fractions

Chapter Practice Test

Solve Equations with a Coefficient of $−1$