Solve Equations with Fractions

4.7 Solve Equations with Fractions

Learning Objectives

By the end of this section, you will be able to:

  • Determine whether a fraction is a solution of an equation
  • Solve equations with fractions using the Addition, Subtraction, and Division Properties of Equality
  • Solve equations using the Multiplication Property of Equality
  • Translate sentences to equations and solve

Be Prepared 4.17

Before you get started, take this readiness quiz. If you miss a problem, go back to the section listed and review the material.

Evaluate x+4x+4 when x=−3x=−3
If you missed this problem, review Example 3.23.

Be Prepared 4.18

Solve: 2y−3=9.2y−3=9.
If you missed this problem, review Example 3.61.

Be Prepared 4.19

Solve: y−3=−9y−3=−9
If you missed this problem, review Example 4.28.

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.

How To

Determine whether a number is a solution to an equation.

  1. Step 1. Substitute the number for the variable in the equation.
  2. Step 2. Simplify the expressions on both sides of the equation.
  3. Step 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.

Example 4.95

Determine whether each of the following is a solution of x−310=12.x−310=12.

  1. ⓐ x=1x=1
  2. ⓑ x=45x=45
  3. ⓒ x=−45x=−45

Solution

ⓐ
A mathematical equation shows 'x minus three tenths equals one half' on a white background. The equation is represented as x - 3/10 = 1/2.
The image displays the mathematical instruction 'Substitute 1 for x.' in a bold, teal-colored font, with the number '1' highlighted in red. Mathematical expression 1 - 3/10 ?= 1/2, questioning the equality. The number '1' is colored red.
Change to fractions with a LCD of 10. A mathematical equation shows '10/10 - 3/10 ? 5/10', where the question mark represents an unknown operator or symbol between the two sides of the equation.
Subtract. A mathematical expression states that 7/10 is not equal to 5/10. The numbers are written in black against a white background.

Since x=1x=1 does not result in a true equation, 11 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. The image displays a mathematical equation: 5/10 = 5/10, followed by a checkmark, indicating that the equality is correct.

Since x=45x=45 results in a true equation, 4545 is a solution to the equation x−310=12.x−310=12.

ⓒ
A mathematical equation shows 'x - 3/10 = 1/2' on a white background, representing an algebraic problem to solve for the variable x.
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. A mathematical expression showing that -11/10 is not equal to 5/10.

Since x=−45x=−45 does not result in a true equation, −45−45 is not a solution to the equation.

Try It 4.189

Determine whether each number is a solution of the given equation.

x−23=16x−23=16:

  1. ⓐ x=1x=1
  2. ⓑ x=56x=56
  3. ⓒ x=−56x=−56

Try It 4.190

Determine whether each number is a solution of the given equation.

y−14=38y−14=38:

  1. ⓐ y=1y=1
  2. ⓑ y=−58y=−58
  3. ⓒ y=58y=58

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,a,b, and c,c,

if a=b,a=b, then a+c=b+c.a+c=b+c. Addition Property of Equality
if a=b,a=b, then a−c=b−c.a−c=b−c. Subtraction Property of Equality
if a=b,a=b, then ac=bc,c≠0.ac=bc,c≠0. Division Property of Equality
Table 4.3 This table lists fundamental properties of equality including addition, subtraction, and division, showing their mathematical statements.

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.

Example 4.96

Solve: y+916=516.y+916=516.

Solution

A mathematical equation is displayed, reading 'y + 9/16 = 5/16'.
Subtract 916916 from each side to undo the addition. An equation showing y + 9/16 - 9/16 = 5/16 - 9/16. The subtracted 9/16 terms are highlighted in red on both sides of the equality, suggesting they might be canceled out.
Simplify on each side of the equation. A mathematical equation shows 'y + 0 = -4/16' centered on a white background, representing a simplified algebraic expression where y is equal to -1/4.
Simplify the fraction. The mathematical equation y = -1/4 is displayed on a white background, representing a horizontal line at y equals negative one-fourth.
Check: A mathematical equation displays 'y + 9/16 = 5/16'.
Substitute y=−14y=−14. 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=−14y=−14 makes y+916=516y+916=516 a true statement, we know we have found the solution to this equation.

Try It 4.191

Solve: y+1112=512.y+1112=512.

Try It 4.192

Solve: y+815=415.y+815=415.

We used the Subtraction Property of Equality in Example 4.96. Now we’ll use the Addition Property of Equality.

Example 4.97

Solve: a−59=−89.a−59=−89.

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 5959 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. A mathematical equation shows 'a + 0 = -3/9' written in black text on a white background.
Simplify the fraction. The image displays a mathematical equation: a = -1/3. The variable 'a' is shown to be equal to negative one-third, presented in a clear, standard mathematical notation against a white background.
Check: A mathematical equation shows 'a minus five ninths equals negative eight ninths'.
Substitute a=−13a=−13. 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. A mathematical equation displays -8/9 = -8/9, accompanied by a checkmark, confirming its correctness.

Since a=−13a=−13 makes the equation true, we know that a=−13a=−13 is the solution to the equation.

Try It 4.193

Solve: a−35=−85.a−35=−85.

Try It 4.194

Solve: n−37=−97.n−37=−97.

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

Example 4.98

Solve: 10q=44.10q=44.

Solution

10q=4410q=44
Divide both sides by 10 to undo the multiplication. 10q10=441010q10=4410
Simplify. q=225q=225
Check:
Substitute q=225q=225 into the original equation. 10(225)=?4410(225)=?44
Simplify. 102(225)=?44102(225)=?44
Multiply. 44=44✓44=44✓
Illustrates the step-by-step process of solving the linear equation 10q=44 and verifying its solution.

The solution to the equation was the fraction 225.225. We leave it as an improper fraction.

Try It 4.195

Solve: 12u=−76.12u=−76.

Try It 4.196

Solve: 8m=92.8m=92.

Solve Equations with Fractions Using the Multiplication Property of Equality

Consider the equation x4=3.x4=3. We want to know what number divided by 44 gives 3.3. So to “undo” the division, we will need to multiply by 4.4. 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.

The Multiplication Property of Equality

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

ifa=b,thenac=bc.ifa=b,thenac=bc.

If you multiply both sides of an equation by the same quantity, you still have equality.

Let’s use the Multiplication Property of Equality to solve the equation x7=−9.x7=−9.

Example 4.99

Solve: x7=−9.x7=−9.

Solution

A mathematical equation is displayed on a white background, reading 'x/7 = -9'.
Use the Multiplication Property of Equality to multiply both sides by 77. This will isolate the variable. An algebraic step where both sides of the equation are multiplied by 7 to isolate 'x', shown as 7 * (x/7) = 7(-9).
Multiply. A mathematical equation is displayed on a white background: 7x/7 = -63.
Simplify. A mathematical equation displays 'x = -63' in black text against a white background.
The text reads 'Check.Substitute -63 for x for in the original equation.' with '-63' highlighted in red, indicating a step in solving an algebraic problem. 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. The mathematical equation -9 = -9 is shown, followed by a checkmark, indicating that the equality is correct.

Try It 4.197

Solve: f5=−25.f5=−25.

Try It 4.198

Solve: h9=−27.h9=−27.

Example 4.100

Solve: p−8=−40.p−8=−40.

Solution

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

The mathematical equation p over -8 equals -40 is displayed on a white background.
Multiply both sides by −8−8 A mathematical equation shows '-8' multiplied by 'P divided by -8' on the left side, which equals '-8' multiplied by '-40' on the right side. The -8 coefficients are highlighted in red.
Multiply. A mathematical equation shows a fraction with '-8p' in the numerator and '-8' in the denominator, set equal to '320'.
Simplify. The image displays the mathematical equation p = 320 in black text against a white background.
Check:
Substitute p=320p=320. A mathematical equation shows '320' in red, divided by '-8', with a question mark over an equals sign, followed by '-40'. It asks whether 320 divided by -8 is equal to -40.
The equation is true. A mathematical equation shows '-40 = -40' with a black checkmark symbol to its right, indicating that the equality is correct. The numbers and symbols are in a dark gray font against a white background.

Try It 4.199

Solve: c−7=−35.c−7=−35.

Try It 4.200

Solve: x−11=−12.x−11=−12.

Solve Equations with a Coefficient of −1−1

Look at the equation −y=15.−y=15. Does it look as if yy is already isolated? But there is a negative sign in front of y,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 4.101.

Example 4.101

Solve: −y=15.−y=15.

Solution

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

The image displays the equation -y = 15, representing a simple linear algebraic expression.
Rewrite −y−y as −1y−1y. A mathematical equation is displayed, showing -1y = 15.
Divide both sides by −1. The equation shows -1y divided by -1 equals 15 divided by -1, demonstrating a step in solving for y by dividing both sides by -1.
Simplify each side. The image displays the equation y = -15, rendered in a clear, standard mathematical font on a white background.

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

The image shows a mathematical equation with a variable 'y'. The equation reads '-y = 15'.
Multiply both sides by −1. A mathematical equation shows '-1(-y) = -1(15)', where both sides are multiplied by -1. The -1 is highlighted in red on both sides.
Simplify each side. The image displays a mathematical equation in black text on a white background, stating 'y = -15'.

The third way to solve the equation is to read −y−y as “the opposite of yy.” What number has 1515 as its opposite? The opposite of 1515 is −15.−15. So y=−15.y=−15.

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

Check:

The image shows a mathematical equation with a variable 'y'. The equation reads '-y = 15'.
Substitute y=−15y=−15. A mathematical equation shows -(-15) ?= (15), asking if the negative of negative 15 is equal to 15, with the -15 in red text.
Simplify. The equation is true. The number 15 is equal to 15, confirmed by a checkmark indicating correctness or verification.

Try It 4.201

Solve: −y=48.−y=48.

Try It 4.202

Solve: −c=−23.−c=−23.

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 1.1.

For example, in the equation:

34x=2434x=24

The coefficient of xx is 34.34. To solve for x,x, we need its coefficient to be 1.1. Since the product of a number and its reciprocal is 1,1, our strategy here will be to isolate xx by multiplying by the reciprocal of 34.34. We will do this in Example 4.102.

Example 4.102

Solve: 34x=24.34x=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. The mathematical expression 'x = 32' is displayed on a white background.
Check: A mathematical equation is displayed on a white background, which reads '3/4x = 24'.
Substitute x=32x=32. A mathematical equation asks if three-fourths multiplied by thirty-two equals twenty-four, with a question mark over the equals sign to indicate inquiry.
Rewrite 3232 as a fraction. A mathematical equation shows (3/4) multiplied by (32/1) with a question mark over the equality sign, followed by the number 24, suggesting a check to see if the product equals 24.
Multiply. The equation is true. A mathematical statement showing 24 = 24, followed by a checkmark to indicate correctness or verification.

Notice that in the equation 34x=24,34x=24, we could have divided both sides by 3434 to get xx 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.

Try It 4.203

Solve: 25n=14.25n=14.

Try It 4.204

Solve: 56y=15.56y=15.

Example 4.103

Solve: −38w=72.−38w=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.

A mathematical equation showing negative three-eighths multiplied by 'w' equals 72.
Multiply both sides by the reciprocal of −38−38. 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. A mathematical equation is displayed, showing the variable 'w' equal to the negative integer -192, presented in a clear, dark font against a stark white background.
Check: A mathematical equation displays '-3/8 w = 72' in black text against a white background.
Let w=−192w=−192. A mathematical expression showing negative three-eighths multiplied by negative one hundred ninety-two, followed by an equals sign with a question mark above it, and then the number seventy-two.
Multiply. It checks. The equation 72=72 is marked as correct with a checkmark.

Try It 4.205

Solve: −47a=52.−47a=52.

Try It 4.206

Solve: −79w=84.−79w=84.

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,a, b, and c,c,

if a=b,a=b, then a−c=b−c.a−c=b−c.
Addition Property of Equality:
For any real numbers a, b,a, b, and c,c,

if a=b,a=b, then a+c=b+c.a+c=b+c.
Division Property of Equality:
For any numbers a, b,a, b, and c,c, where c≠0c≠0

if a=b,a=b, then ac=bcac=bc
Multiplication Property of Equality:
For any real numbers a, b,a, b, and cc

if a=b,a=b, then ac=bcac=bc

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.

Example 4.104

Translate and solve: nn divided by 66 is −24.−24.

Solution

Translate. The image shows the phrase 'n divided by 6 is 24' with brackets indicating its parts. Below, the equation 'n/6 = -24' is displayed, translating the phrase but with a changed sign for the number 24.
Multiply both sides by 66. The equation 6 times n/6 equals 6 times -24, illustrating a step in solving for 'n'.
Simplify. The image displays a mathematical equation in a black serif font on a white background, which states 'n = -144'.
Check: Is −144−144 divided by 66 equal to −24−24?
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. A mathematical equation displays '-24 = -24' followed by a checkmark, confirming the equality.

Try It 4.207

Translate and solve: nn divided by 77 is equal to −21.−21.

Try It 4.208

Translate and solve: nn divided by 88 is equal to −56.−56.

Example 4.105

Translate and solve: The quotient of qq and −5−5 is 70.70.

Solution

Translate. The image shows the verbal expression 'The quotient of q and -5 is 70' translated into the mathematical equation q/-5 = 70. Brackets link corresponding parts of the phrase and the equation.
Multiply both sides by −5−5. A mathematical equation shows -5 multiplied by q over -5, which equals -5 multiplied by 70.
Simplify. The mathematical equation 'q = -350' is displayed on a white background, representing a variable 'q' equal to the negative integer three hundred fifty.
Check: Is the quotient of −350−350 and −5−5 equal to 7070?
Translate. A mathematical problem demonstrating the division of negative numbers, asking if -350 divided by -5 equals 70, symbolized by a question mark over an equals sign between the two values.
Simplify. It checks. The equation 70 = 70 is displayed, followed by a black checkmark, indicating correctness or verification.

Try It 4.209

Translate and solve: The quotient of qq and −8−8 is 72.72.

Try It 4.210

Translate and solve: The quotient of pp and −9−9 is 81.81.

Example 4.106

Translate and solve: Two-thirds of ff is 18.18.

Solution

Translate. A mathematical expression shows how the phrase 'Two-thirds of f is 18' translates into the algebraic equation '(2/3)f = 18,' with curly brackets indicating the corresponding parts.
Multiply both sides by 3232. 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. The image displays the equation 'f = 27' in black text on a white background.
Check: Is two-thirds of 2727 equal to 1818?
Translate. A mathematical equation shows two-thirds multiplied by twenty-seven, followed by a question mark over an equals sign, then eighteen. The equation asks if 2/3(27) is equal to 18.
Simplify. It checks. The image displays the equation '18 = 18' followed by a checkmark, indicating that the mathematical statement is correct.

Try It 4.211

Translate and solve: Two-fifths of ff is 16.16.

Try It 4.212

Translate and solve: Three-fourths of ff is 21.21.

Example 4.107

Translate and solve: The quotient of mm and 5656 is 34.34.

Solution

The quotient of mm and 5656 is 3434.
Translate. m56=34m56=34
Multiply both sides by 5656 to isolate mm. 56(m56)=56(34)56(m56)=56(34)
Simplify. m=5·36·4m=5·36·4
Remove common factors and multiply. m=58m=58
Check:
Is the quotient of 5858 and 5656 equal to 3434? 5856=?345856=?34
Rewrite as division. 58÷56=?3458÷56=?34
Multiply the first fraction by the reciprocal of the second. 58·65=?3458·65=?34
Simplify. 34=34✓34=34✓
This table demonstrates the step-by-step process of solving and verifying an algebraic equation involving fractions for the variable 'm'.

Our solution checks.

Try It 4.213

Translate and solve. The quotient of nn and 2323 is 512.512.

Try It 4.214

Translate and solve The quotient of cc and 3838 is 49.49.

Example 4.108

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

Solution

Translate. This image translates the phrase 'The sum of three-eighths and x is three and one-half' into the algebraic equation 3/8 + x = 3 1/2, illustrating how to set up an equation from a word problem.
Use the Subtraction Property of Equality to subtract 3838 from both sides. A mathematical equation is shown with the expression '3/8 + x - 3/8 = 3 1/2 - 3/8' on a white background, requiring the solving for the variable 'x'.
Combine like terms on the left side. A mathematical equation shows 'x = 3 1/2 - 3/8' centered on a white background, representing a mixed number subtraction problem.
Convert mixed number to improper fraction. A mathematical equation displays 'x = 7/2 - 3/8' centered on a white background.
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. The equation x = 25/8 is displayed on a white background, representing a mathematical solution.
Write as a mixed number. A mathematical expression displays 'x = 3 1/8' on a white background.

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

Check:

Is the sum of three-eighths and 318318 equal to three and one-half?

38+318=?31238+318=?312
Add. 348=?312348=?312
Simplify. 312=312✓312=312✓
Step-by-step verification of a fractional addition problem.

The solution checks.

Try It 4.215

Translate and solve: The sum of five-eighths and xx is one-fourth.

Try It 4.216

Translate and solve: The difference of one-and-three-fourths and xx is five-sixths.

Section 4.7 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.

498.

x−25=110x−25=110:

  1. ⓐ x=1x=1
  2. ⓑ x=12x=12
  3. ⓒ x=−12x=−12
499.

y−13=512y−13=512:

  1. ⓐ y=1y=1
  2. ⓑ y=34y=34
  3. ⓒ y=−34y=−34
500.

h+34=25h+34=25:

  1. ⓐ h=1h=1
  2. ⓑ h=720h=720
  3. ⓒ h=−720h=−720
501.

k+25=56k+25=56:

  1. ⓐ k=1k=1
  2. ⓑ k=1330k=1330
  3. ⓒ k=−1330k=−1330

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

In the following exercises, solve.

502.

y + 1 3 = 4 3 y + 1 3 = 4 3

503.

m + 3 8 = 7 8 m + 3 8 = 7 8

504.

f + 9 10 = 2 5 f + 9 10 = 2 5

505.

h + 5 6 = 1 6 h + 5 6 = 1 6

506.

a − 5 8 = − 7 8 a − 5 8 = − 7 8

507.

c − 1 4 = − 5 4 c − 1 4 = − 5 4

508.

x − ( − 3 20 ) = − 11 20 x − ( − 3 20 ) = − 11 20

509.

z − ( − 5 12 ) = − 7 12 z − ( − 5 12 ) = − 7 12

510.

n − 1 6 = 3 4 n − 1 6 = 3 4

511.

p − 3 10 = 5 8 p − 3 10 = 5 8

512.

s + ( − 1 2 ) = − 8 9 s + ( − 1 2 ) = − 8 9

513.

k + ( − 1 3 ) = − 4 5 k + ( − 1 3 ) = − 4 5

514.

5 j = 17 5 j = 17

515.

7 k = 18 7 k = 18

516.

−4 w = 26 −4 w = 26

517.

−9 v = 33 −9 v = 33

Solve Equations with Fractions Using the Multiplication Property of Equality

In the following exercises, solve.

518.

f 4 = −20 f 4 = −20

519.

b 3 = −9 b 3 = −9

520.

y 7 = −21 y 7 = −21

521.

x 8 = −32 x 8 = −32

522.

p −5 = −40 p −5 = −40

523.

q −4 = −40 q −4 = −40

524.

r −12 = −6 r −12 = −6

525.

s −15 = −3 s −15 = −3

526.

− x = 23 − x = 23

527.

− y = 42 − y = 42

528.

− h = − 5 12 − h = − 5 12

529.

− k = − 17 20 − k = − 17 20

530.

4 5 n = 20 4 5 n = 20

531.

3 10 p = 30 3 10 p = 30

532.

3 8 q = −48 3 8 q = −48

533.

5 2 m = −40 5 2 m = −40

534.

− 2 9 a = 16 − 2 9 a = 16

535.

− 3 7 b = 9 − 3 7 b = 9

536.

− 6 11 u = −24 − 6 11 u = −24

537.

− 5 12 v = −15 − 5 12 v = −15

Mixed Practice

In the following exercises, solve.

538.

3 x = 0 3 x = 0

539.

8 y = 0 8 y = 0

540.

4 f = 4 5 4 f = 4 5

541.

7 g = 7 9 7 g = 7 9

542.

p + 2 3 = 1 12 p + 2 3 = 1 12

543.

q + 5 6 = 1 12 q + 5 6 = 1 12

544.

7 8 m = 1 10 7 8 m = 1 10

545.

1 4 n = 7 10 1 4 n = 7 10

546.

− 2 5 = x + 3 4 − 2 5 = x + 3 4

547.

− 2 3 = y + 3 8 − 2 3 = y + 3 8

548.

11 20 = - f 11 20 = - f

549.

8 15 = - d 8 15 = - d

Translate Sentences to Equations and Solve

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

550.

nn divided by eight is −16.−16.

551.

nn divided by six is −24.−24.

552.

mm divided by −9−9 is −7.−7.

553.

mm divided by −7−7 is −8.−8.

554.

The quotient of ff and −3−3 is −18.−18.

555.

The quotient of ff and −4−4 is −20.−20.

556.

The quotient of gg and twelve is 8.8.

557.

The quotient of gg and nine is 14.14.

558.

Three-fourths of qq is 12.12.

559.

Two-fifths of qq is 20.20.

560.

Seven-tenths of pp is −63.−63.

561.

Four-ninths of pp is −28.−28.

562.

mm divided by 44 equals negative 6.6.

563.

The quotient of hh and 22 is 43.43.

564.

Three-fourths of zz is 15.15.

565.

The quotient of aa and 2323 is 34.34.

566.

The sum of five-sixths and xx is 12.12.

567.

The sum of three-fourths and xx is 18.18.

568.

The difference of yy and one-fourth is −18.−18.

569.

The difference of yy and one-third is −16.−16.

Everyday Math

570.

Shopping Teresa bought a pair of shoes on sale for $48$48. The sale price was 2323 of the regular price. Find the regular price of the shoes by solving the equation 23p=4823p=48

571.

Playhouse The table in a child’s playhouse is 3535 of an adult-size table. The playhouse table is 1818 inches high. Find the height of an adult-size table by solving the equation 35h=18.35h=18.

Writing Exercises

572.

Example 4.100 describes three methods to solve the equation −y=15.−y=15. Which method do you prefer? Why?

573.

Richard thinks the solution to the equation 34x=2434x=24 is 16.16. Explain why Richard is wrong.

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?