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Mathematics » Introducing Fractions » Multiply and Divide Fractions

# Dividing Fractions

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## Dividing Fractions

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, $$2÷\frac{1}{4}=8.$$

The U.S. coin called a quarter is worth one-fourth of a dollar.

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

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

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.

## 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. $$\frac{3}{4}$$