## Identifying and Using Fraction Operations

Contents

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.

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

**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.

**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:

- \(-\frac{1}{4}+\frac{1}{6}\)
- \(-\frac{1}{4}÷\frac{1}{6}\)

### 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. | |

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:

- \(\frac{5}{x}-\frac{3}{10}\)
- \(\frac{5}{x}·\frac{3}{10}\)

### 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. | \(\frac{x}{4}\) |