Multiplying Fractions

The objective of this lesson is that you learn how to multiply fractions correctly

Numerator and Denominator

The number or algebraic expression that appears on the top line of a fraction is called the numerator of the fraction.

The number of algebraic expression that appears on the bottom line of a fraction is called the denominator of the fraction.

Adding Fractions

Expressed in symbols, the rule for multiplying two fractions is as follows:

That is, you simply multiply the two numerators together to form the numerator of the product.

Then you multiply the two denominators together to form the denominator of the product.

Example

Work out each of the following products of fractions.

Solution

(a)

Often it will be possible for you to simplify your fractional expressions by canceling common factors, such as canceling “2” from 56 and 90. This is not strictly necessary, but can sometimes be helpful if it produces a simpler fraction for you to work with.

(b)

In Example (b), note how when the numerators are multiplied every part of the quantity (7·x + 4) is multiplied by 3. It isn’t just the 7·x or the 4 that is multiplied by the 3, it is every part of the entire quantity (7·x + 4). Observe that when the denominators are multiplied together, the same observation holds true: The entire quantity (x + 1) is multiplied by 10.

(c)

When multiplying out the denominator, note that the two quantities (x + 1) and (x - 2) must be multiplied together. Whenever you multiply two quantities in this fashion, you will need to FOIL, just as if you were expanding a quadratic formula that had been written in factored form.

(d) 

This answer is not the simplest one that is possible. If you look closely at the middle fraction above, you can see that every single term in the numerator has at least one factor of (x + 1). The denominator also has a factor of (x + 1). These “common” factors can be factored out of the numerator and the denominator as shown below. 

provided x ≠ -1.

When you have a common factor that you have pulled out of every term in the numerator, and it matches a factor that shows up in the denominator, you can almost always cancel this factor from both the numerator and the denominator.

The only situation when it is not okay to cancel the factor of (x + 1) from the top and bottom is when you have the x-value of x = -1 (i.e. the particular x-value that makes the factor of (x + 1) equal to zero).

The fraction can be further simplified by canceling the common factor of x² (which is permissible when x ≠ 0). Doing this gives: 

provided x ≠ -1 and x ≠ 0.