5.5: Add and Subtract Fractions with Common Denominators
- Page ID
- 41900
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- Model fraction addition
- Add fractions with a common denominator
- Model fraction subtraction
- Subtract fractions with a common denominator
Be Prepared 4.9
Before you get started, take this readiness quiz.
Simplify: 2x+9+3x−4.2x+9+3x−4.
If you missed this problem, review Example 2.22.
Be Prepared 4.10
Draw a model of the fraction 34.34.
If you missed this problem, review Example 4.2.
Be Prepared 4.11
Simplify: 3+26.3+26.
If you missed this problem, review Example 4.48.
Model Fraction Addition
How many quarters are pictured? One quarter plus 22 quarters equals 33 quarters.
Remember, quarters are really fractions of a dollar. Quarters are another way to say fourths. So the picture of the coins shows that
142434one quarter+two quarters=three quarters142434one quarter+two quarters=three quarters
Let’s use fraction circles to model the same example, 14+24.14+24.
Start with one 1414 piece. | ||
Add two more 1414pieces. | ||
The result is 3434. |
So again, we see that
14+24=3414+24=34
Manipulative Mathematics
Example 4.52
Use a model to find the sum 38+28.38+28.
- Answer
Start with three 1818 pieces. Add two 1818pieces. How many 1818pieces are there? There are five 1818 pieces, or five-eighths. The model shows that 38+28=58.38+28=58.
Try It 4.103
Use a model to find each sum. Show a diagram to illustrate your model.
18+4818+48
Try It 4.104
Use a model to find each sum. Show a diagram to illustrate your model.
16+4616+46
Add Fractions with a Common Denominator
Example 4.52 shows that to add the same-size pieces—meaning that the fractions have the same denominator—we just add the number of pieces.
Fraction Addition
If a,b,a,b, and cc are numbers where c≠0,c≠0, then
ac+bc=a+bcac+bc=a+bc
To add fractions with a common denominator, add the numerators and place the sum over the common denominator.
Example 4.53
Find the sum: 35+15.35+15.
- Answer
35+1535+15 Add the numerators and place the sum over the common denominator. 3+153+15 Simplify. 4545
Try It 4.105
Find each sum: 36+26.36+26.
Try It 4.106
Find each sum: 310+710.310+710.
Example 4.54
Find the sum: x3+23.x3+23.
- Answer
x3+23x3+23 Add the numerators and place the sum over the common denominator. x+23x+23 Note that we cannot simplify this fraction any more. Since xx and 22 are not like terms, we cannot combine them.
Try It 4.107
Find the sum: x4+34.x4+34.
Try It 4.108
Find the sum: y8+58.y8+58.
Example 4.55
Find the sum: −9d+3d.−9d+3d.
- Answer
We will begin by rewriting the first fraction with the negative sign in the numerator.
− a b = − a b − a b = − a b
−9d+3d−9d+3d Rewrite the first fraction with the negative in the numerator. −9d+3d−9d+3d Add the numerators and place the sum over the common denominator. −9+3d−9+3d Simplify the numerator. −6d−6d Rewrite with negative sign in front of the fraction. −6d−6d
Try It 4.109
Find the sum: −7d+8d.−7d+8d.
Try It 4.110
Find the sum: −6m+9m.−6m+9m.
Example 4.56
Find the sum: 2n11+5n11.2n11+5n11.
- Answer
2n11+5n112n11+5n11 Add the numerators and place the sum over the common denominator. 2n+5n112n+5n11 Combine like terms. 7n117n11
Try It 4.111
Find the sum: 3p8+6p8.3p8+6p8.
Try It 4.112
Find the sum: 2q5+7q5.2q5+7q5.
Example 4.57
Find the sum: −312+(−512).−312+(−512).
- Answer
−312+(−512)−312+(−512) Add the numerators and place the sum over the common denominator. −3+(−5)12−3+(−5)12 Add. −812−812 Simplify the fraction. −23−23
Try It 4.113
Find each sum: −415+(−615).−415+(−615).
Try It 4.114
Find each sum: −521+(−921).−521+(−921).
Model Fraction Subtraction
Subtracting two fractions with common denominators is much like adding fractions. Think of a pizza that was cut into 1212 slices. Suppose five pieces are eaten for dinner. This means that, after dinner, there are seven pieces (or 712712 of the pizza) left in the box. If Leonardo eats 22 of these remaining pieces (or 212212 of the pizza), how much is left? There would be 55 pieces left (or 512512 of the pizza).
712−212=512712−212=512
Let’s use fraction circles to model the same example, 712−212.712−212.
Start with seven 112112 pieces. Take away two 112112 pieces. How many twelfths are left?
Again, we have five twelfths, 512.512.
Manipulative Mathematics
Example 4.58
Use fraction circles to find the difference: 45−15.45−15.
- Answer
Start with four 1515 pieces. Take away one 1515 piece. Count how many fifths are left. There are three 1515 pieces left.
Try It 4.115
Use a model to find each difference. Show a diagram to illustrate your model.
78−4878−48
Try It 4.116
Use a model to find each difference. Show a diagram to illustrate your model.
56−4656−46
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.
Fraction Subtraction
If a,b,a,b, and cc are numbers where c≠0,c≠0, then
ac−bc=a−bcac−bc=a−bc
To subtract fractions with a common denominator, we subtract the numerators and place the difference over the common denominator.
Example 4.59
Find the difference: 2324−1424.2324−1424.
- Answer
2324−14242324−1424 Subtract the numerators and place the difference over the common denominator. 23−142423−1424 Simplify the numerator. 924924 Simplify the fraction by removing common factors. 3838
Try It 4.117
Find the difference: 1928−728.1928−728.
Try It 4.118
Find the difference: 2732−1132.2732−1132.
Example 4.60
Find the difference: y6−16.y6−16.
- Answer
y6−16y6−16 Subtract the numerators and place the difference over the common denominator. y−16y−16 The fraction is simplified because we cannot combine the terms in the numerator.
Try It 4.119
Find the difference: x7−27.x7−27.
Try It 4.120
Find the difference: y14−1314.y14−1314.
Example 4.61
Find the difference: −10x−4x.−10x−4x.
- Answer
Remember, the fraction −10x−10x can be written as −10x.−10x.
−10x−4x−10x−4x Subtract the numerators. −10−4x−10−4x Simplify. −14x−14x Rewrite with the negative sign in front of the fraction. −14x−14x
Try It 4.121
Find the difference: −9x−7x.−9x−7x.
Try It 4.122
Find the difference: −17a−5a.−17a−5a.
Now lets do an example that involves both addition and subtraction.
Example 4.62
Simplify: 38+(−58)−18.38+(−58)−18.
- Answer
38+(−58)−1838+(−58)−18 Combine the numerators over the common denominator. 3+(−5)−183+(−5)−18 Simplify the numerator, working left to right. −2−18−2−18 Subtract the terms in the numerator. −38−38 Rewrite with the negative sign in front of the fraction. −38−38
Try It 4.123
Simplify: 25+(−45)−35.25+(−45)−35.
Try It 4.124
Simplify: 59+(−49)−79.59+(−49)−79.
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Section 4.4 Exercises
Practice Makes Perfect
Model Fraction Addition
In the following exercises, use a model to add the fractions. Show a diagram to illustrate your model.
2 5 + 1 5 2 5 + 1 5
3 10 + 4 10 3 10 + 4 10
1 6 + 3 6 1 6 + 3 6
3 8 + 3 8 3 8 + 3 8
Add Fractions with a Common Denominator
In the following exercises, find each sum.
4 9 + 1 9 4 9 + 1 9
2 9 + 5 9 2 9 + 5 9
6 13 + 7 13 6 13 + 7 13
9 15 + 7 15 9 15 + 7 15
x 4 + 3 4 x 4 + 3 4
y 3 + 2 3 y 3 + 2 3
7 p + 9 p 7 p + 9 p
8 q + 6 q 8 q + 6 q
8 b 9 + 3 b 9 8 b 9 + 3 b 9
5 a 7 + 4 a 7 5 a 7 + 4 a 7
−12 y 8 + 3 y 8 −12 y 8 + 3 y 8
−11 x 5 + 7 x 5 −11 x 5 + 7 x 5
− 1 8 + ( − 3 8 ) − 1 8 + ( − 3 8 )
− 1 8 + ( − 5 8 ) − 1 8 + ( − 5 8 )
− 3 16 + ( − 7 16 ) − 3 16 + ( − 7 16 )
− 5 16 + ( − 9 16 ) − 5 16 + ( − 9 16 )
− 8 17 + 15 17 − 8 17 + 15 17
− 9 19 + 17 19 − 9 19 + 17 19
6 13 + ( − 10 13 ) + ( − 12 13 ) 6 13 + ( − 10 13 ) + ( − 12 13 )
5 12 + ( − 7 12 ) + ( − 11 12 ) 5 12 + ( − 7 12 ) + ( − 11 12 )
Model Fraction Subtraction
In the following exercises, use a model to subtract the fractions. Show a diagram to illustrate your model.
5 8 − 2 8 5 8 − 2 8
5 6 − 2 6 5 6 − 2 6
Subtract Fractions with a Common Denominator
In the following exercises, find the difference.
4 5 − 1 5 4 5 − 1 5
4 5 − 3 5 4 5 − 3 5
11 15 − 7 15 11 15 − 7 15
9 13 − 4 13 9 13 − 4 13
11 12 − 5 12 11 12 − 5 12
7 12 − 5 12 7 12 − 5 12
4 21 − 19 21 4 21 − 19 21
− 8 9 − 16 9 − 8 9 − 16 9
y 17 − 9 17 y 17 − 9 17
x 19 − 8 19 x 19 − 8 19
5 y 8 − 7 8 5 y 8 − 7 8
11 z 13 − 8 13 11 z 13 − 8 13
− 8 d − 3 d − 8 d − 3 d
− 7 c − 7 c − 7 c − 7 c
− 23 u − 15 u − 23 u − 15 u
− 29 v − 26 v − 29 v − 26 v
6 c 7 − 5 c 7 6 c 7 − 5 c 7
12 d 11 − 9 d 11 12 d 11 − 9 d 11
−4 r 13 − 5 r 13 −4 r 13 − 5 r 13
−7 s 3 − 7 s 3 −7 s 3 − 7 s 3
− 3 5 − ( − 4 5 ) − 3 5 − ( − 4 5 )
− 3 7 − ( − 5 7 ) − 3 7 − ( − 5 7 )
− 7 9 − ( − 5 9 ) − 7 9 − ( − 5 9 )
− 8 11 − ( − 5 11 ) − 8 11 − ( − 5 11 )
Mixed Practice
In the following exercises, perform the indicated operation and write your answers in simplified form.
− 5 18 · 9 10 − 5 18 · 9 10
− 3 14 · 7 12 − 3 14 · 7 12
n 5 − 4 5 n 5 − 4 5
6 11 − s 11 6 11 − s 11
− 7 24 + 2 24 − 7 24 + 2 24
− 5 18 + 1 18 − 5 18 + 1 18
8 15 ÷ 12 5 8 15 ÷ 12 5
7 12 ÷ 9 28 7 12 ÷ 9 28
Everyday Math
Trail Mix Jacob is mixing together nuts and raisins to make trail mix. He has 610610 of a pound of nuts and 310310 of a pound of raisins. How much trail mix can he make?
Baking Janet needs 5858 of a cup of flour for a recipe she is making. She only has 3838 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
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: 116116, 1818, ___, ___, 516516, 3838, ___, 1212, 916916, 5858.
Bits that fell out: 716716, 316316, 1414.
After a party, Lupe has 512512 of a cheese pizza, 412412 of a pepperoni pizza, and 412412 of a veggie pizza left. Will all the slices fit into 11 pizza box? Explain your reasoning.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ 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?