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Mathematics » Introducing Fractions » Add and Subtract Fractions with Different Denominators

# Identifying and Using Fraction Operations

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## Identifying and Using Fraction Operations

By now in this tutorial, you have practiced multiplying, dividing, adding, and subtracting fractions. The following table summarizes these four fraction operations. Remember: You need a common denominator to add or subtract fractions, but not to multiply or divide fractions

### Summary of Fraction Operations

Fraction multiplication: Multiply the numerators and multiply the denominators.

$$\cfrac{a}{b}·\cfrac{c}{d}=\cfrac{ac}{bd}$$

Fraction division: Multiply the first fraction by the reciprocal of the second.

$$\cfrac{a}{b}÷\cfrac{c}{d}=\cfrac{a}{b}·\cfrac{d}{c}$$

Fraction addition: Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.

$$\cfrac{a}{c}+\cfrac{b}{c}=\cfrac{a+b}{c}$$

Fraction subtraction: Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.

$$\cfrac{a}{c}-\cfrac{b}{c}=\cfrac{a-b}{c}$$

## Example

Simplify:

1. $$-\frac{1}{4}+\frac{1}{6}$$
2. $$-\frac{1}{4}÷\frac{1}{6}$$

### Solution

First we ask ourselves, “What is the operation?”

Do the fractions have a common denominator? No.

 $$-\frac{1}{4}+\frac{1}{6}$$ Find the LCD. Rewrite each fraction as an equivalent fraction with the LCD. Simplify the numerators and denominators. $$-\frac{3}{12}+\frac{2}{12}$$ Add the numerators and place the sum over the common denominator. $$-\frac{1}{12}$$ Check to see if the answer can be simplified. It cannot.

The operation is division. We do not need a common denominator.

 $$-\frac{1}{4}÷\frac{1}{6}$$ To divide fractions, multiply the first fraction by the reciprocal of the second. $$-\frac{1}{4}·\frac{6}{1}$$ Multiply. $$-\frac{6}{4}$$ Simplify. $$-\frac{3}{2}$$

## Example

Simplify:

1. $$\frac{5}{x}-\frac{3}{10}$$
2. $$\frac{5}{x}·\frac{3}{10}$$

### Solution

The operation is subtraction. The fractions do not have a common denominator.

 $$\frac{5x}{6}-\frac{3}{10}$$ Rewrite each fraction as an equivalent fraction with the LCD, 30. $$\frac{5x·5}{6·5}-\frac{3·3}{10·3}$$ $$\frac{25x}{30}-\frac{9}{30}$$ Subtract the numerators and place the difference over the common denominator. $$\frac{25x-9}{30}$$

The operation is multiplication; no need for a common denominator.

 $$\frac{5x}{6}·\frac{3}{10}$$ To multiply fractions, multiply the numerators and multiply the denominators. $$\frac{5x·3}{6·10}$$ Rewrite, showing common factors. $$\require{cancel}\frac{\cancel{5}·x·\cancel{3}}{2·\cancel{3}·2·\cancel{5}}$$ Remove common factors to simplify. $$\frac{x}{4}$$