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4.13: Solve Equations with Fractions (Part 2)

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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:

34x=24

The coefficient of x is 34. 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 34. We will do this in Example 4.13.1.

Example 4.13.8: solve

Solve: 34x=24.

Solution

Multiply both sides by the reciprocal of the coefficient. 4334x=4324
Simplify. 1x=43241
Multiply. x=32

Check:

Substitute x = 32. 3432?=24
Rewrite 32 as a fraction. 34321?=24
Multiply. The equation is true. 24=24

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

Exercise 4.13.15

Solve: 25n=14.

Answer

35

Exercise 4.13.16

Solve: 56y=15.

Answer

18

Example 4.13.9: solve

Solve: 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.

Multiply both sides by the reciprocal of 38. 83(38w)=(83)72
Simplify; reciprocals multiply to one. 1w=83721
Multiply. w=192

Check:

Let w = −192. 38(192)?=72
Multiply. It checks. 72=72
Exercise 4.13.17

Solve: 47a=52.

Answer

91

Exercise 4.13.18

Solve: 79w=84.

Answer

108

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.

Table 4.13.2
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 ≠ 0 if a = b, then ac=bc. Multiplication Property of Equality: For any real numbers a, b, and c if a = b, then ac = 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 4.13.10: solve

Translate and solve: n divided by 6 is 24.

Solution

Translate. CNX_BMath_Figure_04_07_037_img-01.png
Multiply both sides by 6. 6n6=6(24)
Simplify. n=144
Check: Is −144 divided by 6 equal to −24?
Translate. 1446?=24
Simplify. It checks. 24=24
Exercise 4.13.19

Translate and solve: n divided by 7 is equal to 21.

Answer

n7=21; n=147

Exercise 4.13.20

Translate and solve: n divided by 8 is equal to 56.

Answer

n8=56; n=448

Example 4.13.11: solve

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

Solution

Translate. CNX_BMath_Figure_04_07_038_img-01.png
Multiply both sides by −5. 5(q5)=5(70)
Simplify. q=350
Check: Is the quotient of −350 and −5 equal to 70?
Translate. 3505?=70
Simplify. It checks. 70=70
Exercise 4.13.21

Translate and solve: The quotient of q and 8 is 72.

Answer

q8=72; q=576

Exercise 4.13.22

Translate and solve: The quotient of p and 9 is 81.

Answer

p9=81; p=729

Example 4.13.12: solve

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

Solution

Translate. CNX_BMath_Figure_04_07_039_img-01.png
Multiply both sides by 32. 3223f=3218
Simplify. f=27
Check: Is two-thirds of 27 equal to 18?
Translate. 23(27)?=18
Simplify. It checks. 18=18
Exercise 4.13.23

Translate and solve: Two-fifths of f is 16.

Answer

25f=16; f=40

Exercise 4.13.24

Translate and solve: Three-fourths of f is 21.

Answer

34f=21; f=28

Example 4.13.13: solve

Translate and solve: The quotient of m and 56 is 34.

Solution

Translate. m56=34
Multiply both sides by frac56 to isolate m. 56(m56)=56(34)
Simplify. m=5364
Remove common factors and multiply. m=58

Check:

Is the quotient of 58 and 56 equal to 34? 5856?=34
Rewrite as division. 58÷56?=34
Multiply the first fraction by the reciprocal of the second. 5865?=34
Simplify. 34=34

Our solution checks.

Exercise 4.13.25

Translate and solve. The quotient of n and 23 is 512.

Answer

n23=512; n=518

Exercise 4.13.26

Translate and solve. The quotient of c and 38 is 49.

Answer

c38=49; c=16

Example 4.13.14: solve

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

Solution

Translate. CNX_BMath_Figure_04_07_040_img-01.png
Use the Subtraction Property of Equality to subtract 38 from both sides. 38+x38=31238
Combine like terms on the left side. x=31238
Convert mixed number to improper fraction. x=31238
Convert to equivalent fractions with LCD of 8. x=7238
Subtract. x=258
Write as a mixed number. x=318

We write the answer as a mixed number because the original problem used a mixed number. Check: Is the sum of three-eighths and 318 equal to three and one-half?

Add. 348?=312
Simplify. 312=312

The solution checks.

Exercise 4.13.27

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

Answer

58+x=14; x=38

Exercise 4.13.28

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

Answer

134x=56; x=1112

Access Additional Online Resources

  • Solve One Step Equations With Fractions
  • Solve One Step Equations With Fractions by Adding or Subtracting
  • Solve One Step Equations With Fraction by Multiplying

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.

  1. x − 25 = 110:
    1. x = 1
    2. x = 12
    3. x = 12
  2. y − 12 = 512:
    1. y = 1
    2. y = 34
    3. y = 34
  3. h + 34 = 25:
    1. h = 1
    2. h = 720
    3. h = 720
  4. k + 25 = 56:
    1. k = 1
    2. k = 1330
    3. k = 1330

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

In the following exercises, solve.

  1. y + 13 = 43
  2. m + 38 = 78
  3. f + 910 = 25
  4. h + 56 = 16
  5. a − 58 = 78
  6. c − 14 = 54
  7. x − (320) = 1120
  8. z − (512) = 712
  9. n − 16 = 34
  10. p − 310 = 58
  11. s + (12) = 89
  12. k + (13) = 45
  13. 5j = 17
  14. 7k = 18
  15. −4w = 26
  16. −9v = 33

Solve Equations with Fractions Using the Multiplication Property of Equality

In the following exercises, solve.

  1. f4 = −20
  2. b3 = −9
  3. y7 = −21
  4. x8 = −32
  5. p5 = −40
  6. q4 = −40
  7. r12 = −6
  8. s15 = −3
  9. −x = 23
  10. −y = 42
  11. −h = 512
  12. −k = 1720
  13. 45n = 20
  14. 310p = 30
  15. 38q = −48
  16. 52m = −40
  17. 29a = 16
  18. 37b = 9
  19. 611u = −24
  20. 512v = −15

Mixed Practice

In the following exercises, solve.

  1. 3x = 0
  2. 8y = 0
  3. 4f = 45
  4. 7g = 79
  5. p + 23 = 112
  6. q + 56 = 112
  7. 78m = 110
  8. 14n = 710
  9. 25 = x + 34
  10. 23 = y + 38
  11. 1120 = −f
  12. 815 = −d

Translate Sentences to Equations and Solve

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

  1. n divided by eight is −16.
  2. n divided by six is −24.
  3. m divided by −9 is −7.
  4. m divided by −7 is −8.
  5. The quotient of f and −3 is −18.
  6. The quotient of f and −4 is −20.
  7. The quotient of g and twelve is 8.
  8. The quotient of g and nine is 14.
  9. Three-fourths of q is 12.
  10. Two-fifths of q is 20.
  11. Seven-tenths of p is −63.
  12. Four-ninths of p is −28.
  13. m divided by 4 equals negative 6.
  14. The quotient of h and 2 is 43.
  15. Three-fourths of z is the same as 15.
  16. The quotient of a and 23 is 34.
  17. The sum of five-sixths and x is 12.
  18. The sum of three-fourths and x is 18.
  19. The difference of y and one-fourth is 18.
  20. The difference of y and one-third is 16.

Everyday Math

  1. Shopping Teresa bought a pair of shoes on sale for $48. The sale price was 23 of the regular price. Find the regular price of the shoes by solving the equation 23p = 48
  2. Playhouse The table in a child’s playhouse is 35 of an adult-size table. The playhouse table is 18 inches high. Find the height of an adult-size table by solving the equation 35h = 18.

Writing Exercises

  1. Example 4.100 describes three methods to solve the equation −y = 15. Which method do you prefer? Why?
  2. Richard thinks the solution to the equation 34x = 24 is 16. Explain why Richard is wrong.

Self Check

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

CNX_BMath_Figure_AppB_026.jpg

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

Contributors and Attributions


This page titled 4.13: Solve Equations with Fractions (Part 2) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax.

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