# Adding and Subtracting Fractions with Like Denominators

Alice and Bob order a cake which has been cut into six equal slices. Thus, each slice is of the whole cake. Alice eats two slices (shaded in light green in Figure below), or of the whole cake. Tony eats three slices (shaded in light red in the figure below), or of the whole cake.

Now it should be clear that together Alice and Bob eat ﬁve slices or of the whole cake. This reﬂects the fact that:

This demonstrates how to add two fractions with like (same) denominator. Keep the common denominator and add the numerators. That is,

### Note:

Let and be two fractions with like (same) denominator. Their sum is deﬁned as:

Subtracting Fractions with Like Denominators

A similar rule holds for subtraction.

### Note:

Let and be two fractions with like (same) denominator. Their difference is deﬁned as:

## Examples of Adding and Subtracting Fractions with Like Denominator

#### Example1:

Add the fractions and reduce to the lowest terms if possible:

#### Solution:

Both denominators are the same. So, we can add the numerators 2 and 5, and keep the denominator which is 3.

Th fraction is in the lowest terms because

#### Example2:

Add the fractions and reduce to the lowest terms if possible:

#### Solution:

Both denominators are the same. So, we can add the numerators 5 and 1, and keep the denominator which is 12.

Th fraction isn't in the lowest terms because .

#### Example3:

Add the fractions and reduce to the lowest terms if possible:

#### Solution:

Both denominators are the same. So, we can add the numerators 2 and 5, and keep the denominator which is 7.

Th fraction isn't in the lowest terms because .

#### Example4:

Add the fractions and reduce to the lowest terms if possible:

#### Solution:

Both denominators are the same. So, we can add the numerators 2 and -7, and keep the denominator which is 9.

Th fraction is in the lowest terms because

#### Example5:

Add the fractions and reduce to the lowest terms if possible:

#### Solution:

Both denominators are the same. So, we can add the numerators -2 and -8, and keep the denominator which is 5.

Th fraction isn't in the lowest terms because

#### Example6:

Subtract the fractions and reduce to the lowest terms if possible:

#### Solution:

Both denominators are the same. So, we can subtract the numerators 8 and 3, and keep the denominator which is 11.

Th fraction is in the lowest terms because

#### Example7:

Subtract the fractions and reduce to the lowest terms if possible:

#### Solution:

Both denominators are the same. So, we can subtract the numerators 7 and -3, and keep the denominator which is 16.

Th fraction isn't in the lowest terms because

#### Example8:

Subtract the fractions and reduce to the lowest terms if possible:

#### Solution:

Both denominators are the same. So, we can subtract the numerators -11 and 9, and keep the denominator which is 21.

Th fraction is in the lowest terms because

#### Example9:

Add the fractions and reduce to the lowest terms if possible:

#### Solution:

Both denominators are the same. So, we can add the numerators 2 and 5, and keep the denominator which is -13.

Th fraction is in the lowest terms because

#### Example10:

Add the fractions and reduce to the lowest terms if possible:

#### Solution:

Both denominators aren't the same(4 and -4), but we can make them with the same denominator. We multiply the denominator and numerator of the fraction by -1, we get:

Now, we add the fractions with like denominators.

Th fraction is in the lowest terms because

#### Example11:

Add the fractions and reduce to the lowest terms if possible:

#### Solution:

The previous property can be extended as a sum of three fractions.

### Note:

Let , and be three fractions with like (same) denominator. Their sum is deﬁned as:

Both denominators are the same. So, we can add the numerators 2, 5 and 7, and keep the denominator which is 3.

Th fraction is in the lowest terms because

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