Mathematics » Introducing Fractions » Multiply and Divide Fractions

# Multiplying Fractions

## Multiplying Fractions

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

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

## 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