Find the sum: 3 2

7 7+

Find the diffe rence:6 4

7 7-

Adding or subtracting fractions

with the same denominator

Add or subtract the numerators.

Write the result over the common denominator.

Find the sum: 7 6

11 11+

Find the difference: 13 7

8 8-

Find the sum: 4

15

+

Adding or subtracting fractions

with the different denominators

Rewrite the sum or difference as the sum or difference of fractions with a common denominator.

Find the sum: 1 1

2 4+

Find the sum: 1 1

2 4+

The common denominator 4, used in the previous example, is called the lowest common denominator.

The number 4 is the least common multiple of 2 and 4.

Lowest common denominator

Least common multiple

Multiples of 2: 2, 4, 6, 8, 10, …

Multiples of 4: 4, 8, 12, 16, …

The least common multiple of two or more numbers is the smallest number that each divides into evenly.

The least common multiple of two or more numbers is often abbreviated as LCM.

Least common multiple

Lowest common denominator

The lowest common denominator of two or more fractions is the least common multiple of all the denominators.

The lowest common denominator of two or more fractions is often abbreviated as LCD.

2Find the sum:

3

1

4+

2Find the sum:

3

1

4+

1Find the sum:

6

3

4+

and 1 1 2 1 1 3

,2 4 3 4 6 4

+ + +

What is similar? What is different?

1 1 2 1 1 3

2 4 3 4 6 4+ + +

Similarities

In the first example, 2 and 4 share a factor of 2, and the denominator 4 is the lowest common denominator.

In the third example, 6 and 4 share a common factor of 2, but neither denominator is the lowest common denominator.

Differences

In the second example, the two denominators share no common factors except 1; the lowest common denominator is the product of the two denominators.

Find the difference: 11 1

15 6-

Find the prime factors of each denominator.

Find the difference: 11 1

15 6-

Strategy for finding the least common multiple of two or more numbers.

Find the prime factorization of each number.

List all primes that occur in one or more factorizations.

For each prime factor, identify the maximum number of times it occurs in any one factorization. This is the number of occurrences in the least common multiple.

Strategy for finding the least common multiple of two or more numbers.

Find the least common multiple of 30 and 45.

Adding or subtracting fractions with different denominators

The goal is to rewrite the problem so

that the fractions have a common

denominator.

Factor each denominator into prime

factors.

Multiply each fraction in the sum or

difference by the appropriate form of

1 to determine the equivalent fraction

with the common denominator.

Find the least common multiple of all

the denominators to use as the

common denominator:

Adding or subtracting fractions with different denominators

The appropriate form of 1 is found by

determining the prime factors needed

to complete the common

denominator.

Add or subtract the equivalent

fractions with the common

denominator.

Adding or subtracting fractions with different denominators

5Evaluate

12

2.9

-

3Evaluate

14

1 2.

10 15+

In this part of the lesson, and in the lesson on multiplying and dividing fractions, we rewrote numerators and/or denominators as the product of prime factors.

The following shortcut is useful when performing arithmetic problems, but the strategies we discussed in these lessons will be the only strategy to use when working with algebraic fractions.

Shortcuts

Evaluate 1 2 1

.4 5 10

+ +

Shortcuts