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Mathematics LibreTexts

3.6: Add and Subtract Fractions with Common Denominators

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Learning Objectives
  • Model fraction addition
  • Add fractions with a common denominator
  • Model fraction subtraction
  • Subtract fractions with a common denominator
be prepared!

Before you get started, take this readiness quiz.

  1. Simplify: 2x+9+3x4. If you missed this problem, review Example 2.2.10.
  2. Draw a model of the fraction 34. If you missed this problem, review Example 4.1.2.
  3. Simplify: 3+26. If you missed this problem, review Example 4.3.12.

Model Fraction Addition

How many quarters are pictured? One quarter plus 2 quarters equals 3 quarters.

Three U.S. quarters are shown. One is shown on the left, and two are shown on the right.

Figure 3.6.1

Remember, quarters are really fractions of a dollar. Quarters are another way to say fourths. So the picture of the coins shows that

142434onequarter+twoquarters=threequarters

Let’s use fraction circles to model the same example, 14+24.

Start with one 14 piece. CNX_BMath_Figure_04_04_002_img-01.png 14
Add two more 14 pieces. CNX_BMath_Figure_04_04_002_img-02.png +24
The result is 34. CNX_BMath_Figure_04_04_002_img-03.png 34

So again, we see that

14+24=34

Example 3.6.1: addition

Use a model to find the sum 38+28.

Solution

Start with three 18 pieces. CNX_BMath_Figure_04_04_003_img-01.png 38
Add two 18 pieces. CNX_BMath_Figure_04_04_003_img-02.png +28
How many 18 pieces are there? CNX_BMath_Figure_04_04_003_img-03.png 58

There are five 18 pieces, or five-eighths. The model shows that 38+28=58.

Exercise 3.6.1

Use a model to find each sum. Show a diagram to illustrate your model. 18+48

Answer

58

CNX_BMath_Figure_04_04_004_img.jpg

Exercise 3.6.2

Use a model to find each sum. Show a diagram to illustrate your model. 16+46

Answer

56

CNX_BMath_Figure_04_04_005_img.jpg

Add Fractions with a Common Denominator

Example 3.6.1 shows that to add the same-size pieces—meaning that the fractions have the same denominator—we just add the number of pieces.

Definition: Fraction Addition

If a, b, and c are numbers where c0, then

ac+bc=a+bc

To add fractions with a common denominator, add the numerators and place the sum over the common denominator.

Example 3.6.2: addition

Find the sum: 35+15.

Solution

Add the numerators and place the sum over the common denominator. 3+15
Simplify. 45
Exercise 3.6.3

Find each sum: 36+26.

Answer

56

Exercise 3.6.4

Find each sum: 310+710.

Answer

1

Example 3.6.3: addition

Find the sum: x3+23.

Solution

Add the numerators and place the sum over the common denominator. x+23

Note that we cannot simplify this fraction any more. Since x and 2 are not like terms, we cannot combine them.

Exercise 3.6.5

Find the sum: x4+34.

Answer

x+34

Exercise 3.6.6

Find the sum: y8+58.

Answer

y+58

Example 3.6.4: addition

Find the sum: 9d+3d.

Solution

We will begin by rewriting the first fraction with the negative sign in the numerator.

ab=ab

Rewrite the first fraction with the negative in the numerator. 9d+3d
Add the numerators and place the sum over the common denominator. 9+3d
Simplify the numerator. 6d
Rewrite with negative sign in front of the fraction. 6d
Exercise 3.6.7

Find the sum: 7d+8d.

Answer

1d

Exercise 3.6.8

Find the sum: 6m+9m.

Answer

3m

Example 3.6.5: addition

Find the sum: 2n11+5n11.

Solution

Add the numerators and place the sum over the common denominator. 2n+5n11
Combine like terms. 7n11
Exercise 3.6.9

Find the sum: 3p8+6p8.

Answer

9p8

Exercise 3.6.10

Find the sum: 2q5+7q5.

Answer

9q5

Example 3.6.6: addition

Find the sum: 312+(512).

Solution

Add the numerators and place the sum over the common denominator. 3+(5)12
Add. 812
Simplify the fraction. 23
Exercise 3.6.11

Find each sum: 415+(615).

Answer

23

Exercise 3.6.12

Find each sum: 521+(921).

Answer

23

Model Fraction Subtraction

Subtracting two fractions with common denominators is much like adding fractions. Think of a pizza that was cut into 12 slices. Suppose five pieces are eaten for dinner. This means that, after dinner, there are seven pieces (or 712 of the pizza) left in the box. If Leonardo eats 2 of these remaining pieces (or 212 of the pizza), how much is left? There would be 5 pieces left (or 512 of the pizza).

712212=512

Let’s use fraction circles to model the same example, 712212. Start with seven 112 pieces. Take away two 112 pieces. How many twelfths are left?

The bottom reads 7 twelfths minus 2 twelfths equals 5 twelfths. Above 7 twelfths, there is a circle divided into 12 equal pieces, with 7 pieces shaded in orange. Above 2 twelfths, the same circle is shown, but 2 of the 7 pieces are shaded in grey. Above 5 twelfths, the 2 grey pieces are no longer shaded, so there is a circle divided into 12 pieces with 5 of the pieces shaded in orange.

Figure 3.6.2

Again, we have five twelfths, 512.

Example 3.6.7: difference

Use fraction circles to find the difference: 4515.

Solution

Start with four 15 pieces. Take away one 15 piece. Count how many fifths are left. There are three 15 pieces left.

The bottom reads 4 fifths minus 1 fifth equals 3 fifths. Above 4 fifths, there is a circle divided into 5 equal pieces, with 4 pieces shaded in orange. Above 1 fifth, the same circle is shown, but 1 of the 4 shaded pieces is shaded in grey. Above 3 fifths, the 1 grey piece is no longer shaded, so there is a circle divided into 5 pieces with 3 of the pieces shaded in orange.

Exercise 3.6.13

Use a model to find each difference. Show a diagram to illustrate your model. 7848

Answer

38, models may differ.

Exercise 3.6.14

Use a model to find each difference. Show a diagram to illustrate your model. 5646

Answer

16, models may differ.

Subtract Fractions with a Common Denominator

We subtract fractions with a common denominator in much the same way as we add fractions with a common denominator.

Definition: Fraction Subtraction

If a, b, and c are numbers where c0, then

acbc=abc

To subtract fractions with a common denominator, we subtract the numerators and place the difference over the common denominator.

Example 3.6.8: difference

Find the difference: 23241424.

Solution

Subtract the numerators and place the difference over the common denominator. 231424
Simplify the numerator. 924
Simplify the fraction by removing common factors. 38
Exercise 3.6.15

Find the difference: 1928728.

Answer

37

Exercise 3.6.16

Find the difference: 27321132.

Answer

12

Example 3.6.9: difference

Find the difference: y616.

Solution

Subtract the numerators and place the difference over the common denominator. y16

The fraction is simplified because we cannot combine the terms in the numerator.

Exercise 3.6.17

Find the difference: x727.

Answer

x27

Exercise 3.6.18

Find the difference: y141314.

Answer

y1314

Example 3.6.10: difference

Find the difference: 10x4x.

Solution

Remember, the fraction 10x can be written as 10x.

Subtract the numerators. 104x
Simplify. 14x
Rewrite with the negative sign in front of the fraction. 14x
Exercise 3.6.19

Find the difference: 9x7x.

Answer

16x

Exercise 3.6.20

Find the difference: 17a5a.

Answer

22a

Now lets do an example that involves both addition and subtraction.

Example 3.6.11: simplify

Simplify: 38+(58)18.

Solution

Combine the numerators over the common denominator. 3+(5)18
Simplify the numerator, working left to right. 218
Subtract the terms in the numerator. 38
Rewrite with the negative sign in front of the fraction. 38
Exercise 3.6.21

Simplify: 25+(45)35.

Answer

1

Exercise 3.6.22

Simplify: 59+(49)79.

Answer

23

Access Additional Online Resources

  • Adding Fractions With Pattern Blocks
  • Adding Fractions With Like Denominators
  • Subtracting Fractions With Like Denominators

Key Concepts

  • Fraction Addition
    • If a,b, and c are numbers where c0, then ac+bc=a+bc
    • To add fractions, add the numerators and place the sum over the common denominator.
  • Fraction Subtraction
    • If a,b, and c are numbers where c0, then acbc=abc
    • To subtract fractions, subtract the numerators and place the difference over the common denominator.

Practice Makes Perfect

Model Fraction Addition

In the following exercises, use a model to add the fractions. Show a diagram to illustrate your model.

  1. 25+15
  2. 310+410
  3. 16+36
  4. 38+38

Add Fractions with a Common Denominator

In the following exercises, find each sum.

  1. 49+19
  2. 29+59
  3. 613+713
  4. 915+715
  5. x4+34
  6. y3+23
  7. 7p+9p
  8. 8q+6q
  9. 8b9+3b9
  10. 5a7+4a7
  11. 12y8+3y8
  12. 11x5+7x5
  13. 18+(38)
  14. 18+(58)
  15. 316+(716)
  16. 516+(916)
  17. 817+1517
  18. 919+1719
  19. 613+(1013)+(1213)
  20. 512+(712)+(1112)

Model Fraction Subtraction

In the following exercises, use a model to subtract the fractions. Show a diagram to illustrate your model.

  1. 5828
  2. 5626

Subtract Fractions with a Common Denominator

In the following exercises, find the difference.

  1. 4515
  2. 4535
  3. 1115715
  4. 913413
  5. 1112512
  6. 712512
  7. 4211921
  8. 89169
  9. y17917
  10. x19819
  11. 5y878
  12. 11z13813
  13. 8d3d
  14. 7c7c
  15. 23u15u
  16. 29v26v
  17. 6c75c7
  18. 12d119d11
  19. 4r135r13
  20. 7s37s3
  21. 35(45)
  22. 37(57)
  23. 79(59)
  24. 811(511)

Mixed Practice

In the following exercises, perform the indicated operation and write your answers in simplified form.

  1. 518910
  2. 314712
  3. n545
  4. 611s11
  5. 724224
  6. 518118
  7. 815÷125
  8. 712÷928

Everyday Math

  1. Trail Mix Jacob is mixing together nuts and raisins to make trail mix. He has 610 of a pound of nuts and 310 of a pound of raisins. How much trail mix can he make?
  2. Baking Janet needs 58 of a cup of flour for a recipe she is making. She only has 38 of a cup of flour and will ask to borrow the rest from her next-door neighbor. How much flour does she have to borrow?

Writing Exercises

  1. Greg dropped his case of drill bits and three of the bits fell out. The case has slots for the drill bits, and the slots are arranged in order from smallest to largest. Greg needs to put the bits that fell out back in the case in the empty slots. Where do the three bits go? Explain how you know.

Bits in case: 116,18, ___, ___, 516,38, ___, 12,916,58.

Bits that fell out: 716,316,14.

  1. After a party, Lupe has 512 of a cheese pizza, 412 of a pepperoni pizza, and 412 of a veggie pizza left. Will all the slices fit into 1 pizza box? Explain your reasoning.

Self Check

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

CNX_BMath_Figure_AppB_023.jpg

(b) On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

Contributors and Attributions


This page titled 3.6: Add and Subtract Fractions with Common Denominators is shared under a not declared license and was authored, remixed, and/or curated by OpenStax.

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