Multiplying Fractions

Many situations require us to multiply fractions. For instance, suppose that a mixture in a chemistry class calls for g of sodium chloride. If we make only of that mixture, we need of , that is, g of sodium chloride.

To illustrate how to find this product, we diagram these two fractions.

In the following diagram, we are taking of the .

Note that we divided the whole into 15 parts and that our product, containing 8 of the 15 small squares, represents the double-shaded region. The answer is therefore of the original whole, which we can compute as follows.

The numerator and denominator of the answer are the products of the original numerators and denominators.

To Multiply Fractions

• first multiply the numerators,
• then multiply the denominators, and
• finally write the answer in simplest form.

EXAMPLE 1

Multiply: .

Solution

EXAMPLE 2

What is of 10?

Solution

Finding of 10 means multiplying by 10.

In Example 2, we multiplied the two fractions first and then simplified the answer. It is preferable, however, to reverse these steps: Simplify first and then multiply. By first simplifying, which is called canceling, we divide any numerator and any denominator by a common factor. Canceling before multiplying allows us to work with smaller numbers and still gives us the same answer.

EXAMPLE 3

Find the product of .

Solution

EXAMPLE 4

Multiply: .

Solution

We cancel and then multiply.

EXAMPLE 5

At a college, of the students take a math course. Of these students, take elementary algebra. What fraction of the students in the college take elementary algebra?

Solution

We must find of .

One-tenth of the students in the college take elementary algebra.

EXAMPLE 6

Suppose that you spend of your monthly salary on rent. If your salary is $960, how much do you have left after paying the rent?

Solution

Apply the strategy of breaking the question into two parts.

• First, find of $960.
• Then, subtract that result from $960.

Thus you can solve this problem by computing .

You have $600 left after paying the rent.