# Fractions

### Operations With Fractions--Addition and Subtraction

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 . 7 = 7
(b) 42×2 = 84 . 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 . 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

#### 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).

Example: 5 3/4 + 6 5/6 = ?

5 + 6 = 11

3/4 + 5/6 = ?

I. The LCD of 3/4 and 5/6 is 2×2×3 = 12
II. 3/4 = 9/12 and 5/6 = 10/12
III. 9/12 + 10/12 = 19/12.
IV. 19/12 cannot be reduced further
V. As a mixed number, 19/12 = 1 7/12
11 + 1 7/12 = 12 7/12
Thus, 5 3/4 + 6 5/6 = 12 7/12

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