## Multiplying Fractions

Contents

A model may help you understand multiplication of fractions. We will use fraction tiles to model \(\frac{1}{2}·\frac{3}{4}.\) To multiply \(\frac{1}{2}\) and \(\frac{3}{4},\) think \(\frac{1}{2}\) of \(\frac{3}{4}.\)

Start with fraction tiles for three-fourths. To find one-half of three-fourths, we need to divide them into two equal groups. Since we cannot divide the three \(\frac{1}{4}\) tiles evenly into two parts, we exchange them for smaller tiles.

We see \(\frac{6}{8}\) is equivalent to \(\frac{3}{4}.\) Taking half of the six \(\frac{1}{8}\) tiles gives us three \(\frac{1}{8}\) tiles, which is \(\frac{3}{8}.\)

Therefore,

\(\frac{1}{2}·\frac{3}{4}=\frac{3}{8}\)

## Example

Use a diagram to model \(\frac{1}{2}·\frac{3}{4}.\)

### Solution

First shade in \(\frac{3}{4}\) of the rectangle.

We will take \(\frac{1}{2}\) of this \(\frac{3}{4},\) so we heavily shade \(\frac{1}{2}\) of the shaded region.

Notice that \(3\) out of the \(8\) pieces are heavily shaded. This means that \(\frac{3}{8}\) of the rectangle is heavily shaded.

Therefore, \(\frac{1}{2}\) of \(\frac{3}{4}\) is \(\frac{3}{8},\) or \(\frac{1}{2}·\frac{3}{4}=\frac{3}{8}.\)

Look at the result we got from the model in the example above. We found that \(\frac{1}{2}·\frac{3}{4}=\frac{3}{8}.\) Do you notice that we could have gotten the same answer by multiplying the numerators and multiplying the denominators?

\(\frac{1}{2}·\frac{3}{4}\) | |

Multiply the numerators, and multiply the denominators. | \(\frac{1}{2}·\frac{3}{4}\) |

Simplify. | \(\frac{3}{8}\) |

This leads to the definition of fraction multiplication. To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.

### Fraction Multiplication

If \(a,b,c,\text{and}\phantom{\rule{0.2em}{0ex}}d\) are numbers where \(b\ne 0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d\ne 0,\) then

\(\frac{a}{b}·\frac{c}{d}=\frac{ac}{bd}\)

### Optional Video: Multiplying Fractions (Positive Only)

## Example

Multiply, and write the answer in simplified form: \(\frac{3}{4}·\frac{1}{5}.\)

### Solution

\(\frac{3}{4}·\frac{1}{5}\) | |

Multiply the numerators; multiply the denominators. | \(\frac{3·1}{4·5}\) |

Simplify. | \(\frac{3}{20}\) |

There are no common factors, so the fraction is simplified.

When multiplying fractions, the properties of positive and negative numbers still apply. It is a good idea to determine the sign of the product as the first step. In the example below, we will multiply two negatives, so the product will be positive.

## Example

Multiply, and write the answer in simplified form: \(-\frac{5}{8}\left(-\frac{2}{3}\right).\)

### Solution

\(-\frac{5}{8}\left(-\frac{2}{3}\right)\) | |

The signs are the same, so the product is positive. Multiply the numerators, multiply the denominators. | \(\frac{5\cdot 2}{8\cdot 3}\) |

Simplify. | \(\frac{10}{24}\) |

Look for common factors in the numerator and denominator. Rewrite showing common factors. | |

Remove common factors. | \(\frac{5}{12}\) |

Another way to find this product involves removing common factors earlier.

\(-\frac{5}{8}\left(-\frac{2}{3}\right)\) | |

Determine the sign of the product. Multiply. | \(\frac{5\cdot 2}{8\cdot 3}\) |

Show common factors and then remove them. | |

Multiply remaining factors. | \(\frac{5}{12}\) |

We get the same result.

## Example

Multiply, and write the answer in simplified form: \(-\frac{14}{15}·\frac{20}{21}.\)

### Solution

\(-\frac{14}{15}·\frac{20}{21}\) | |

Determine the sign of the product; multiply. | \(-\frac{14}{15}·\frac{20}{21}\) |

Are there any common factors in the numerator and the denominator? \(\phantom{\rule{0.2em}{0ex}}\)We know that 7 is a factor of 14 and 21, and 5 is a factor of 20 and 15. | |

Rewrite showing common factors. | |

Remove the common factors. | \(-\frac{2·4}{3·3}\) |

Multiply the remaining factors. | \(-\frac{8}{9}\) |

When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, \(a,\) can be written as \(\frac{a}{1}.\) So, \(3=\frac{3}{1},\) for example.

## Example

Multiply, and write the answer in simplified form:

\(\frac{1}{7}·56\)

\(\frac{12}{5}\left(-20x\right)\)

### Solution

\(\frac{1}{7}·56\) | |

Write 56 as a fraction. | \(\frac{1}{7}·\frac{56}{1}\) |

Determine the sign of the product; multiply. | \(\frac{56}{7}\) |

Simplify. | \(8\) |

\(\frac{12}{5}\left(-20x\right)\) | |

Write −20x as a fraction. | \(\frac{12}{5}\left(\frac{-20x}{1}\right)\) |

Determine the sign of the product; multiply. | \(-\frac{12·20·x}{5·1}\) |

Show common factors and then remove them. | |

Multiply remaining factors; simplify. | −48x |