Ch 4 Sec 4: Slide #1 Columbus State Community College Chapter 4 Section 4 Adding and Subtracting Signed Fractions

Ch 4 Sec 4: Slide #1

Columbus State Community College

Chapter 4 Section 4

Adding and Subtracting Signed Fractions



1. Add and subtract like fractions.

2. Find the lowest common denominator for unlike fractions.

3. Add and subtract unlike fractions.

4. Add and subtract unlike fractions that contain variables.


Fractions

Like Fractions Unlike Fractions

Common denominator

Common denominator

2m

9m

and

Different denominators

45

49

and

Different denominators

6a

78

and

35

15

and


Adding and Subtracting Like Fractions


You can add or subtract fractions only when they have a common denominator. If a, b, and c are numbers (and b is not 0), then

ab

a + cb

=cb

+ and ab

a – cb

=cb

–

In other words, add or subtract the numerators and write the result over the common denominator. Then check to be sure that the answer is in lowest terms.


3

1


Find each sum or difference.

EXAMPLE 1 Adding and Subtracting Like Fractions

Common denominator

(a) 19

29

+

19

1 + 29

=29

+39

=13

=


Adding Fractions

Add only the numerators. Do not add the denominators. In part (a)

we kept the common denominator.

CAUTION

Incorrect

19

1 + 29

=29

+ not19

1 + 29 + 9

=29

+3

18=





Common denominator

(b) 45

15

+

45 5

=15

+4 + 1

5=

3

5=

3





Common denominator

(c) 27

67

–

27 7

=67

– 2 – 67

=4

7=

4





Common denominator

(d) 3k

2k

+

3k k

=2k

+3 + 2

k=

5


A Common Denominator for Unlike Fractions

To find a common denominator for two unlike fractions, find a

number that is divisible by both of the original denominators.

For example, a common denominator for and is 18

because 6 goes into 18 evenly and 9 goes into 18 evenly.

A Common Denominator for Unlike Fractions

61

91


Least Common Denominator (LCD)

The least common denominator (LCD) for two fractions is the

smallest positive number divisible by both denominators of the

original fractions.

For example, both 8 and 16 are common denominators for

and , but 8 is smaller, so it is the LCD.

Least Common Denominator (LCD)

41

81


Finding the LCD by Inspection

EXAMPLE 2 Finding the LCD by Inspection

Check to see if 14 (the larger denominator) will work as the LCD.

Is 14 divisible by 7 (the other denominator)?

Yes, so 14 is the LCD for and .

(a) Find the LCD for and .57

314

57

314


Finding the LCD by Inspection

EXAMPLE 2 Finding the LCD by Inspection

Check to see if 9 (the larger denominator) will work as the LCD.

Is 9 divisible by 6 (the other denominator)?

No, 9 is not divisible by 6. So start checking numbers that are

multiples of 9, that is, 18, 27, and 36.

Notice that 18 will work because it is divisible by 6 and 9.

The LCD for and is 18.

(b) Find the LCD for and .16

29

16

29


Write 20 and 12 as the product of prime factors. Then use prime

factors in the LCD that “cover” both 20 and 12.

60 is divisible by 20 and by 12, it is the LCD for and .

LCD = 2 • 2 • 3 • 5 = 60

Using Prime Factors to Find the LCD

EXAMPLE 3 Using Prime Factors to Find the LCD

112

(a) What is the LCD for and .1

125

20

520

20 = 2 • 2 • 5

12 = 2 • 2 • 3

Factors of 20

Factors of 12


LCD

When finding the LCD, notice that we did not have to repeat the

factors that 20 and 12 have in common. If we had used all the 2s

and 3s, we would get a common denominator, but not the smallest

one.

CAUTION


Write 15 and 40 as the product of prime factors. Then use prime

factors in the LCD that “cover” both 15 and 40.

120 is divisible by 15 and by 40, it is the LCD for and .

LCD = 2 • 2 • 2 • 3 • 5 = 120

Using Prime Factors to Find the LCD

EXAMPLE 3 Using Prime Factors to Find the LCD

340

(b) What is the LCD for and .3

408

15

815

15 = 3 • 5

40 = 2 • 2 • 2 • 5

Factors of 15

Factors of 40


Adding and Subtracting Unlike Fractions

Step 1 Find the LCD, the smallest number divisible by both denomi-

nators in the problem.

Step 2 Rewrite each original fraction as an equivalent fraction whose

denominator is the LCD.

Step 3 Add or subtract the numerators of the like fractions. Keep the

common denominator.

Step 4 Write the sum or difference in lowest terms.



Step 1 The larger denominator ( 8 ) is the LCD.

Step 2

Step 3 Add the numerators. Write the sum over the denominator.

Step 4 is in lowest terms.


EXAMPLE 4 Adding and Subtracting Unlike Fractions

(a) Find the sum. +14

38

14

38

already has the LCD and28

1 • 24 • 2

= =

38

+14

38

= +28

=58

3 + 28

=

58


Step 1 The LCD is 24.

Step 2

Step 3 Subtract the numerators. Write the difference over the

common denominator.



56

2024

5 • 46 • 4

= =78

2124

7 • 38 • 3

= =

2024

– 78

56

= – 2124

=20 – 21

241

24= –

(b) Find the difference.78

56

–





124

–

(b) Find the difference.78

56

–


Step 1 Use prime factorization to find the LCD.



(c) Find the difference.8

631142

–

LCD = 2 • 3 • 3 • 7 = 126

42 = 2 • 3 • 7

63 = 3 • 3 • 7

Factors of 42

Factors or 63


Step 2

Step 3 Subtract the numerators. Write the difference over the

common denominator.




631142

–

1142

33126

11 • 342 • 3

= =8

6316126

8 • 263 • 2

= =

33126

– 863

1142

= – 16126

17126

=33 – 16

126=


Step 4 is in lowest terms.12617




631142

–


3a + 2b6

Step 1 The LCD is 6.

Step 2

Step 3 Add the numerators. Keep the common denominator.


Adding Unlike Fractions with Variables

EXAMPLE 5 Adding Unlike Fractions with Variables

(a) Find the sum. +b3

a2

3a6

+b3

a2

= +2b6

=3a + 2b

6

b3

2b6

b • 23 • 2

= =a2

3a6

a • 32 • 3

= =


Combining Terms

In the previous problem, we could not add 3a + 2b in the numerator

of the answer because 3a and 2b are not like terms. We could add

3a + 2a or 3b + 2b but not 3a + 2b.

CAUTION

Variable parts match.

Variable parts match.


Step 1 The LCD is 4 • n, or 4n.

Step 2

Step 3 Subtract the numerators. Keep the common denominator.

Step 4 is in lowest terms.mn – 284n

Subtracting Unlike Fractions with Variables

EXAMPLE 5 Subtracting Unlike Fractions with Variables

=mn – 28

4n

7n

284n

7 • 4n • 4

= =m4

mn4n

m • n4 • n

= =

(b) Find the difference.7n

m4

–

mn4n

– 7n

m4

= 284n

–


Common Denominators

NOTE

Notice in Example 5 (b) that we found the LCD for

by multiplying the two denominators. The LCD is 4 • n or 4n.

Multiplying the two denominators will always give you a common denominator, but it may not be the smallest common denominator. Here are more examples.

7n

m4

–

34

25

– If you multiply the denominators, 5 • 4 = 20 and 20 is the LCD.

56

18

+If you multiply the denominators, 8 • 6 = 48 and 48 will work. But you’ll save some time by using the smallest common denominator, which is 24.



Chapter 4 Section 4 – Completed

Written by John T. Wallace

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Ch 4 Sec 4: Slide #1 Columbus State Community College Chapter 4 Section 4 Adding and Subtracting Signed Fractions