### 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 [latex]\Large\frac{1}{2}\cdot \frac{3}{4}[/latex]. To multiply [latex]\Large\frac{1}{2}[/latex] and [latex]\Large\frac{3}{4}[/latex], think [latex]\Large\frac{1}{2}[/latex] of [latex]\Large\frac{3}{4}[/latex].

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 [latex]\Large\frac{1}{4}[/latex] tiles evenly into two parts, we exchange them for smaller tiles.

We see [latex]\Large\frac{6}{8}[/latex] is equivalent to [latex]\Large\frac{3}{4}[/latex]. Taking half of the six [latex]\Large\frac{1}{8}[/latex] tiles gives us three [latex]\Large\frac{1}{8}[/latex] tiles, which is [latex]\Large\frac{3}{8}[/latex].

Therefore,

[latex]\Large\frac{1}{2}\cdot \frac{3}{4}=\frac{3}{8}[/latex]

### Example

Use a diagram to model [latex]\Large\frac{1}{2}\cdot \frac{3}{4}[/latex]

Solution:

First shade in [latex]\Large\frac{3}{4}[/latex] of the rectangle.

We will take [latex]\Large\frac{1}{2}[/latex] of this [latex]\Large\frac{3}{4}[/latex], so we heavily shade [latex]\Large\frac{1}{2}[/latex] of the shaded region.

Notice that [latex]3[/latex] out of the [latex]8[/latex] pieces are heavily shaded. This means that [latex]\Large\frac{3}{8}[/latex] of the rectangle is heavily shaded.

Therefore, [latex]\Large\frac{1}{2}[/latex] of [latex]\Large\frac{3}{4}[/latex] is [latex]\Large\frac{3}{8}[/latex], or [latex]{\Large\frac{1}{2}\cdot \frac{3}{4}}={\Large\frac{3}{8}}[/latex].

### Try it

Use a diagram to model: [latex]\Large\frac{1}{2}\cdot \frac{3}{5}[/latex]

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

[latex]\Large\frac{1}{2}\cdot \frac{3}{4}[/latex] | |

Multiply the numerators, and multiply the denominators. | [latex]\Large\frac{1}{2}\cdot \frac{3}{4}[/latex] |

Simplify. | [latex]\Large\frac{3}{8}[/latex] |

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 [latex]a,b,c,\text{ and }d[/latex] are numbers where [latex]b\ne 0\text{ and }d\ne 0[/latex], then

[latex]\Large\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}[/latex]

### Example

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

### 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: [latex]\Large-\frac{5}{8}\left(-\frac{2}{3}\right)[/latex]

### 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: [latex]\Large-\frac{14}{15}\cdot \frac{20}{21}[/latex]

### 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, [latex]a[/latex], can be written as [latex]\Large\frac{a}{1}[/latex]. So, [latex]3=\Large\frac{3}{1}[/latex], for example.

### example

Multiply, and write the answer in simplified form:

- [latex]\Large{\frac{1}{7}}\normalsize\cdot 56[/latex]
- [latex]\Large{\frac{12}{5}}\normalsize\left(-20x\right)[/latex]

### Try it

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