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Operations With Fractions--Addition and Subtraction

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Adding and Subtracting Fractions

We can only add or subtract fractions when they have the same
denominator. Therefore, the first step in adding or subtracting fractions
is writing them as fractions with the same denominator
(see Reducing Fractions and the Least Common Denominator).
Once the denominators
have been equalized, adding or subtracting the fractions is easy--simply add or
subtract the numerators, while keeping the denominator the same. The
numerator of the answer is this result, and the denominator of the answer is the
common denominator.

It is often useful to write the answer in lowest terms, using the steps learned
in Reducing Fractions and the Least Common Denominator.

*Example 1*: 1/12 + 5/42 = ?

I. Find the LCD.

1. Factor the denominators. 12 = 2×2×3 and 42 = 2×3×7
2. Find the LCM of the denominators. 2×2×3×7 = 84
3. The LCD is 84.

II. Write each fraction as an equivalent fraction with the LCD (84) as the new
denominator.

(a) 12×**7** = 84. 1×**7** = 7

(b) 42×**2** = 84. 5×**2** = 10

Thus,

1/12 = 7/84 and

5/42 = 10/84
III. Add.

7 + 10 = 17
7/84 + 10/84 = 17/84.BR>

IV. Reduce. Since 17 and 84 have no common factors, the
fraction cannot be reduced further.

1/12 + 5/42 = 17/84

*Example 2*: 13/20 - 3/70 = ?

I. Find the LCD

1. 20 = 2×2×5 and 70 = 2×5×7

2. 2×2×5×7 = 140

3. The LCD is 140

II. Write as equivalent fractions with the LCD as the denominator.

(a) 20×**7** = 140. 13×**7** = 91

(b) 70×**2** = 140. 3×**2** = 6

Thus,

13/20 = 91/140 and

3/70 = 6/140
III. Subtract.

91 - 6 = 85
91/140 - 6/140 = 85/140

IV. Reduce.

1. Factor the numerator and the denominator. 85 = 5×17 and 140 = 2×2×5×7

2. Find the GCF. The GCF is 5.BR>
3. Divide. 85/5 = 17 and 140/5 = 28. Thus, 85/140 = 17/28

13/20 - 3/70 = 17/28.

*Example 3*: 9/8 - 5/12 - 2/5 = ?

I and II. As we have already learned,
these three fractions with common denominators are:

9/8 = 135/120

5/12 = 50/120

2/5 = 48/120

III. Subtract.

135 - 50 - 48 = 37
135/120 - 50/120 - 48/120 = 37/120

IV. Reduce. Since 37 and 120 have no common factors, the fraction cannot be
reduced
further.

9/8 - 5/12 - 2/5 = 37/120

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Adding and Subtracting Mixed Numbers

To add and subtract mixed numbers, first add or subtract the whole numbers
and then add or subtract the fractions as above. If the fractional part is
improper, convert it to a mixed number
(see converting mixed fractions).