## Dividing Fractions

Contents

Why is \(12÷3=4?\) We previously modeled this with counters. How many groups of \(3\) counters can be made from a group of \(12\) counters?

There are \(4\) groups of \(3\) counters. In other words, there are four \(3\text{s}\) in \(12.\) So, \(12÷3=4.\)

What about dividing fractions? Suppose we want to find the quotient: \(\frac{1}{2}÷\frac{1}{6}.\) We need to figure out how many \(\frac{1}{6}\text{s}\) there are in \(\frac{1}{2}.\) We can use fraction tiles to model this division. We start by lining up the half and sixth fraction tiles as shown in the figure below. Notice, there are three \(\frac{1}{6}\) tiles in \(\frac{1}{2},\) so \(\frac{1}{2}÷\frac{1}{6}=3.\)

## Example

Model: \(\frac{1}{4}÷\frac{1}{8}.\)

### Solution

We want to determine how many \(\frac{1}{8}\text{s}\) are in \(\frac{1}{4}.\) Start with one \(\frac{1}{4}\) tile. Line up \(\frac{1}{8}\) tiles underneath the \(\frac{1}{4}\) tile.

There are two \(\frac{1}{8}\text{s}\) in \(\frac{1}{4}.\)

So, \(\frac{1}{4}÷\frac{1}{8}=2.\)

## Example

Model: \(2÷\frac{1}{4}.\)

### Solution

We are trying to determine how many \(\frac{1}{4}\text{s}\) there are in \(2.\) We can model this as shown.

Because there are eight \(\frac{1}{4}\text{s}\) in \(2,2÷\frac{1}{4}=8.\)

Let’s use money to model \(2÷\frac{1}{4}\) in another way. We often read \(\frac{1}{4}\) as a ‘quarter’, and we know that a quarter is one-fourth of a dollar as shown in the figure below. So we can think of \(2÷\frac{1}{4}\) as, “How many quarters are there in two dollars?” One dollar is \(4\) quarters, so \(2\) dollars would be \(8\) quarters. So again, \(2÷\frac{1}{4}=8.\)

Using fraction tiles, we showed that \(\frac{1}{2}÷\frac{1}{6}=3.\) Notice that \(\frac{1}{2}·\frac{6}{1}=3\) also. How are \(\frac{1}{6}\) and \(\frac{6}{1}\) related? They are reciprocals. This leads us to the procedure for fraction division.

### Fraction Division

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

To divide fractions, multiply the first fraction by the reciprocal of the second.

We need to say \(b\ne 0,c\ne 0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d\ne 0\) to be sure we don’t divide by zero.

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

## Example

Divide, and write the answer in simplified form: \(\frac{2}{5}÷\left(-\frac{3}{7}\right).\)

### Solution

\(\frac{2}{5}÷\left(-\frac{3}{7}\right)\) | |

Multiply the first fraction by the reciprocal of the second. | \(\frac{2}{5}\left(-\frac{7}{3}\right)\) |

Multiply. The product is negative. | \(-\frac{14}{15}\) |

## Example

Divide, and write the answer in simplified form: \(\frac{2}{3}÷\frac{n}{5}.\)

### Solution

\(\frac{2}{3}÷\frac{n}{5}\) | |

Multiply the first fraction by the reciprocal of the second. | \(\frac{2}{3}·\frac{5}{n}\) |

Multiply. | \(\frac{10}{3n}\) |

## Example

Divide, and write the answer in simplified form: \(-\frac{3}{4}÷\left(-\frac{7}{8}\right).\)

### Solution

\(-\frac{3}{4}÷\left(-\frac{7}{8}\right)\) | |

Multiply the first fraction by the reciprocal of the second. | \(-\frac{3}{4}·\left(-\frac{8}{7}\right)\) |

Multiply. Remember to determine the sign first. | \(\frac{3·8}{4·7}\) |

Rewrite to show common factors. | \(\require{cancel}\frac{3·\cancel{4}·2}{\cancel{4}·7}\) |

Remove common factors and simplify. | \(\frac{6}{7}\) |

## Example

Divide, and write the answer in simplified form: \(\frac{7}{18}÷\frac{14}{27}.\)

### Solution

\(\frac{7}{18}÷\frac{14}{27}\) | |

Multiply the first fraction by the reciprocal of the second. | \(\frac{7}{18}·\frac{27}{14}\) |

Multiply. | \(\frac{7·27}{18·14}\) |

Rewrite showing common factors. | |

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

Simplify. | \(\frac{3}{4}\) |