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4.7: Adding and Subtracting Mixed Fractions

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In this section, we will learn how to add and subtract mixed fractions.

Adding Mixed Fractions

We can use tools we’ve already developed to add two or more mixed fractions.

Example 1

Simplify: 278+134.

Solution

Change the mixed fractions to improper fractions, make equivalent fractions with a common denominator, then add.

278+134=238+74  Change to equivalent fractions.=238+7242  Equivalent fractions with LCD = 8.=238+148  Simplify numerators and denominators.=378  Add numerators over common denominator.

Although this answer is perfectly acceptable, let’s change the answer to a mixed fraction: 37 divided by 8 is 4, with a remainder of 5. Thus,

278+134=458.

Exercise

Simplify: 323+418

Answer

71924

Example 2

Simplify: 314+213.

Solution

Change the mixed fractions to improper fractions, make equivalent fractions with a common denominator, then add.

314+213=134+73  Change to improper fractions.=13343+7434  Equivalent fractions with LCD = 12.=3912+2812  Simplify numerators and denominators.=6712  Add numerators and denominators.

Although this answer is perfectly acceptable, let’s change the answer to a mixed fraction: 67 divided by 12 is 5, with a remainder of 7. Thus,

314+213=5712.

Exercise

Simplify: 812+223

Answer

1116

Mixed Fraction Approach

There is another possible approach, based on the fact that a mixed fraction is a sum. Let’s revisit Example 2.

Example 3

Simplify: 314+213.

Solution

Use the commutative and associative properties to change the order of addition, make equivalent fractions with a common denominator, then add.

314+213=(3+14)+(2+13)  Mixed fractions as sums.=(3+2)+(14+13)  Reorder and regroup.=5+(1343+1434)  Add whole numbers: 3 + 2 = 5. Equivalent fractions; LCD = 12.=5+(312+412)  Simplify numerators and denominators.=5+712  Add numerators over common denominators.

This result can be written in mixed fraction form. Thus,

314+213=5712.

Note that this solution is identical to the result found in Example 2.

Exercise

Simplify: 725+318

Answer

102140

Example 3 leads us to the following result.

Adding Mixed Fractions

To add two mixed fractions, add the whole number parts, then add the fractional parts.

Working in Vertical Format

When adding mixed fractions, many prefer to work in a vertical format. For example, here is how we would arrange the solution from Example 2 and Example 3 in vertical format. We create equivalent fractions, then add the whole number parts and fractional parts.

314=31343=3312+213=+21434=+2412    5712

Note that the answer is identical to the answer found in Example 2 and Example 3. That is,

314+213=5712.

Example 4

Sarah is making window curtains for two rooms in her house. The kitchen will require 523 yards of material and the dining room will require 658 yards of material. How much total material is required?

Solution

To find the total material required for the two rooms, we must add 523 and 658. Create equivalent fractions with a common denominator, then add whole number parts and fractional parts.

523=52838=51624+658=+65383=+61524113124

An answer that is part mixed fraction, part improper fraction, is not allowed. To finish, we need to change the improper fractional part to a mixed fraction, then add. 31 divided by 24 is 1, with a remainder of 7. That is, 31/24 = 1 7 24 . Now we can add whole number parts and fractional parts.

113124=11+1724=12724.

Thus, the total material required is 12724 yards.

Exercise

Jim is working on a project that requires two boards, the first cut to a length of 612 feet, the second cut to a length of 578 feet. How many total feet of board is required?

Answer

1238 feet.

Subtracting Mixed Fractions

Let’s look at some examples that subtract two mixed fractions.

Example 5

Simplify: 4582116.

Solution

Change the mixed fractions to improper fractions, make equivalent fractions with a common denominator, then subtract.

4582116=3783316  Change to improper fractions.=372823316  Equivalent fractions with LCD = 16.=74163316  Simplify numerator and denominators.=4116  Add numerators over common denominator.

Although this answer is perfectly acceptable, let’s change the answer to a mixed fraction: 41 divided by 16 is 2, with a remainder of 9. Thus,

4582116=2916.

Exercise

Simplify: 523315

Answer

2715

Example 6

Simplify: 534213.

Solution

Change the mixed fractions to improper fractions, make equivalent fractions with a common denominator, then subtract.

534213=23473  Change to improper fractions.=233437434  Equivalent fractions with LCD = 12.=69122812  Simplify numerators and denominators.=4112  Add numerators over common denominator.

Although this answer is perfectly acceptable, let’s change the answer to a mixed fraction: 41 divided by 12 is 3, with a remainder of 5. Thus,

534213=3512.

Exercise

Simplify: 4792318

Answer

21118

Mixed Fraction Approach

There is another possible approach, based on the fact that a mixed fraction is a sum. Let’s revisit Example 6.

Example 7

Simplify: 534213.

Solution

A mixed fraction is a sum.

534213=(5+34)(2+13)

Distribute the negative sign.

=5+34213

We could change the subtraction to adding the opposite, change the order of addition, then change the adding of opposites back to subtraction. However, it is much easier if we look at this last line as a request to add four numbers, two of which are positive and two of which are negative. Changing the order does not affect the answer.

=(52)+(3413)

Note that we did not change the signs of any of the four numbers. We just changed the order. Subtract the whole number parts. Make equivalent fractions with a common denominator, then subtract the fractional parts.

=3+(33431434)  Create equivalent fractions.=3+(912412)  Simplify numerators and denominators.=3+512  Subtract fractional parts.

Thus,

534213=3512.

Note that this is exactly the same answer as that found in Example 6.

Exercise

Simplify: 856438

Answer

41124

In Example 6, we see that we handle subtraction of mixed fractions in exactly the same manner that we handle addition of mixed fractions.

Subtracting Mixed Fractions

To subtract two mixed fractions, subtract their whole number parts, then subtract their fractional parts.

Working in Vertical Format

When subtracting mixed fractions, many prefer to work in a vertical format. For example, here is how we would arrange the solution from Example 6 and Example 7 in vertical format. We create equivalent fractions, then subtract the whole number parts and fractional parts.

534=53343=5912213=31434=24123512

Note that the answer is identical to the answer found in Example 6 and Example 7. That is,

534213=3512.

Borrowing in Vertical Format

Consider the following example.

Example 8

Simplify: 814556.

Solution

Create equivalent fractions with a common denominator.

814=81343=8312556=55262=51012

You can see the difficulty. On the far right, we cannot subtract 10/12 from 3/12. The fix is to borrow 1 from 8 in the form of 12/12 and add it to the 3/12.

8312=7+1212+312=7151251012=51012=510122512

Now we can subtract. Hence, 814556=2512.

Exercise

Simplify: 71142521

Answer

456.

Example 9

Jim has a metal rod of length 10 inches. He cuts a length from the metal rod measuring 278 inches. What is the length of the remaining piece?

Solution

To find the length of the remaining piece, we must subtract 278 from 10. There is no fractional part on the first number. To remedy this absence, we borrow 1 from 10 in the form of 8/8. Then we can subtract.

10=9+88=988278=278=278718

Hence, the length of the remaining piece of the metal rod is 718 inches.

Exercise

Sarah has a length of curtain material that measures 12 feet. She cuts a length of 623 feet from her curtain material. What is the length of the remaining piece?

Answer

513 feet

Exercises

In Exercises 1-24, add or subtract the mixed fractions, as indicated, by first converting each mixed fraction to an improper fraction. Express your answer as a mixed fraction.

1. 914+912

2. 213+912

3. 612113

4. 513134

5. 912+714

6. 113+934

7. 523+412

8. 1916+234

9. 313114

10. 212114

11. 812113

12. 512123

13. 412118

14. 212113

15. 478+134

16. 118+512

17. 213114

18. 513114

19. 912134

20. 5121316

21. 423+114

22. 114+113

23. 912+318

24. 114+123


In Exercises 25-48, add or subtract the mixed fractions, as indicated, by using vertical format. Express your answer as a mixed fraction.

25. 312+334

26. 112+223

27. 138+114

28. 214+123

29. 178+112

30. 134+412

31. 812523

32. 812123

33. 7121316

34. 512113

35. 912113

36. 2121316

37. 513212

38. 414112

39. 912223

40. 712423

41. 1116+134

42. 114+113

43. 812+323

44. 123+212

45. 6121316

46. 412113

47. 223+114

48. 112+1116


Answers

1. 1834

3. 516

5. 1634

7. 1016

9. 2112

11. 716

13. 338

15. 658

17. 1112

19. 734

21. 51112

23. 1258

25. 714

27. 258

29. 338

31. 256

33. 6516

35. 816

37. 256

39. 656

41. 21316

43. 1216

45. 5516

47. 31112


This page titled 4.7: Adding and Subtracting Mixed Fractions is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by David Arnold.

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