## 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.

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.

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.

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: $\Large\frac{1}{2}\cdot \frac{3}{5}$

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: $\Large\frac{3}{4}\cdot \frac{1}{5}$

### 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: $\Large-\frac{5}{8}\left(-\frac{2}{3}\right)$

### 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: $\Large-\frac{14}{15}\cdot \frac{20}{21}$

### 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:

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

### Try it

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