Multiplying Fractions

Learning Outcomes

Use a diagram to model multiplication of positive and negative fractions
Multiply fractions and integer expressions that contain variables

A model may help you understand multiplication of fractions. We will use fraction tiles to model $\Large\frac{1}{2}\cdot \frac{3}{4}$ . To multiply $\Large\frac{1}{2}$ and $\Large\frac{3}{4}$ , think $\Large\frac{1}{2}$ of $\Large\frac{3}{4}$ .
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 $\Large\frac{1}{4}$ tiles evenly into two parts, we exchange them for smaller tiles.

A rectangle is divided vertically into three equal pieces. Each piece is labeled as one fourth. There is a an arrow pointing to an identical rectangle divided vertically into six equal pieces. Each piece is labeled as one eighth. There are braces showing that three of these rectangles represent three eighths.
We see $\Large\frac{6}{8}$ is equivalent to $\Large\frac{3}{4}$ . Taking half of the six $\Large\frac{1}{8}$ tiles gives us three $\Large\frac{1}{8}$ tiles, which is $\Large\frac{3}{8}$ .

Therefore,

$\Large\frac{1}{2}\cdot \frac{3}{4}=\frac{3}{8}$

Example

Use a diagram to model $\Large\frac{1}{2}\cdot \frac{3}{4}$

Solution:
First shade in $\Large\frac{3}{4}$ of the rectangle.

A rectangle is shown, divided vertically into four equal pieces. Three of the pieces are shaded.
We will take $\Large\frac{1}{2}$ of this $\Large\frac{3}{4}$ , so we heavily shade $\Large\frac{1}{2}$ of the shaded region.

A rectangle is shown, divided vertically into four equal pieces. Three of the pieces are shaded. The rectangle is divided by a horizontal line, creating eight equal pieces. Three of the eight pieces are darkly shaded.
Notice that $3$ out of the $8$ pieces are heavily shaded. This means that $\Large\frac{3}{8}$ of the rectangle is heavily shaded.
Therefore, $\Large\frac{1}{2}$ of $\Large\frac{3}{4}$ is $\Large\frac{3}{8}$ , or ${\Large\frac{1}{2}\cdot \frac{3}{4}}={\Large\frac{3}{8}}$ .

Try it

Use a diagram to model: $\Large\frac{1}{2}\cdot \frac{3}{5}$

Show Solution

$\Large\frac{3}{10}$

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

	$\Large\frac{1}{2}\cdot \frac{3}{4}$
Multiply the numerators, and multiply the denominators.	$\Large\frac{1}{2}\cdot \frac{3}{4}$
Simplify.	$\Large\frac{3}{8}$

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 $a,b,c,\text{ and }d$ are numbers where $b\ne 0\text{ and }d\ne 0$ , then

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

Example

Multiply, and write the answer in simplified form: $\Large\frac{3}{4}\cdot \frac{1}{5}$

Show Solution

Solution:

$\Large\frac{3}{4}\cdot \frac{1}{5}$
Multiply the numerators; multiply the denominators.	$\Large\frac{3\cdot 1}{4\cdot 5}$
Simplify.	$\Large\frac{3}{20}$

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

Try It

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 next example, we will multiply two negatives, so the product will be positive.

Example

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

Show Solution

Solution:

	$\Large-\frac{5}{8}\left(-\frac{2}{3}\right)$
The signs are the same, so the product is positive. Multiply the numerators, multiply the denominators.	$\Large\frac{5\cdot 2}{8\cdot 3}$
Simplify.	$\Large\frac{10}{24}$
Look for common factors in the numerator and denominator. Rewrite showing common factors.	$\Large\frac{5\cdot\color{red}{2}}{12\cdot\color{red}{2}}$
Remove common factors.	$\Large\frac{5}{12}$

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

	$\Large-\frac{5}{8}\left(-\frac{2}{3}\right)$
Determine the sign of the product. Multiply.	$\Large\frac{5\cdot 2}{8\cdot 3}$
Show common factors and then remove them.	$\Large\frac{5\cdot\color{red}{2}}{4\cdot\color{red}{2}\cdot3}$
Multiply remaining factors.	$\Large\frac{5}{12}$

We get the same result.

Try it

The following video provides more examples of how to multiply fractions, and simplify the result.

Example

Multiply, and write the answer in simplified form: $\Large-\frac{14}{15}\cdot \frac{20}{21}$

Show Solution

Solution:

	$\Large-\frac{14}{15}\cdot \frac{20}{21}$
Determine the sign of the product; multiply.	$\Large-\frac{14}{15}\cdot \frac{20}{21}$
Are there any common factors in the numerator and the denominator? We know that $7$ is a factor of $14$ and $21$ , and $5$ is a factor of $20$ and $15$ .
Rewrite showing common factors.	$\Large-\frac{2\cdot\color{red}{7}\cdot4\cdot\color{blue}{5}}{3\cdot\color{blue}{5}\cdot3\cdot\color{red}{7}}$
Remove the common factors.	$\Large-\frac{2\cdot 4}{3\cdot 3}$
Multiply the remaining factors.	$\Large-\frac{8}{9}$

Try it

The following video shows another example of multiplying fractions that are negative.

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 $\Large\frac{a}{1}$ . So, $3=\Large\frac{3}{1}$ , for example.

example

Multiply, and write the answer in simplified form:

$\Large{\frac{1}{7}}\normalsize\cdot 56$
$\Large{\frac{12}{5}}\normalsize\left(-20x\right)$

Show Solution

Solution:

1.
	$\Large\frac{1}{7}\normalsize\cdot 56$
Write $56$ as a fraction.	$\Large\frac{1}{7}\cdot \frac{56}{1}$
Determine the sign of the product; multiply.	$\Large\frac{56}{7}$
Simplify.	$8$

2.
	$\Large\frac{12}{5}\normalsize\left(-20x\right)$
Write $−20x$ as a fraction.	$\Large\frac{12}{5}\left(\frac{-20x}{1}\right)$
Determine the sign of the product; multiply.	$\Large-\frac{12\cdot 20\cdot x}{5\cdot 1}$
Show common factors and then remove them.	$\Large-\frac{12\cdot 4\cdot {\color{red}{5}}\cdot x}{\color{red}{5}\cdot1}$
Multiply remaining factors; simplify.	$−48x$

Try it

Watch the following video to see more examples of how to multiply a fraction and a whole number.

Module 3: Fractions