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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?”

The operation is addition.

Do the fractions have a common denominator? No.

 \(-\frac{1}{4}+\frac{1}{6}\)
Find the LCD.

 

Identifying and Using Fraction Operations

 
Rewrite each fraction as an equivalent fraction with the LCD.Identifying and Using Fraction Operations
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.\(\frac{x}{4}\)

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