Adding, Subtracting, Multiplying, and Dividing Fractions Study Guide: GED Math

Multiplying and Dividing Fractions

You do not need to find a common denominator when multiplying or dividing fractions. In this sense, multiplying and dividing fractions is easier than adding and subtracting them. If you know how to multiply, then you basically already know how to multiply and divide fractions.

To multiply fractions, multiply the numerators and then multiply the denominators. Finally, simplify if needed.

Example

Multiply the numerators and denominators:

Simplify your result: .

You can simplify your multiplication by canceling before multiplying. Like reducing a fraction, canceling involves dividing. If you can see a number that will divide evenly into one of the numerators and one of the denominators of each fraction you are multiplying, then do so. This is canceling.

Example

There is a 3 in the numerator of the first fraction and in the denominator of the second fraction. You can cancel by dividing by 3:

You have simplified your problem to . Do you see a way to further simply this problem with canceling? Both 4 and 2 can be divided by 2:

So now your multiplication is very easy:

.

Dividing fractions is very similar to multiplying fractions. To divide a fraction by another fraction, follow these steps.

Step 1   Invert the second fraction. That is, write the numerator on the bottom and the denominator on the top. The new fraction is the reciprocal of the original fraction.

Step 2   Multiply the two fractions.

Step 3   Write the answer in lowest terms.

Example

Find the reciprocal of .

Multiply the first fraction by the reciprocal of the second: .

Multiply the numerators and denominators:

Simplify your result: .

Improper Fractions

So far, the fractions you have been working with have all been proper fractions. A proper fraction is one in which the numerator is smaller than the denominator. These are examples of proper fractions: , and so on. Proper fractions are always equal to or less than 1. They represent a part of whole.

The numerator of an improper fraction is the same as or greater than its denominator. Here are some examples of improper fractions: and so on. Remember that the bar in a fraction means to divide the top number by the bottom number. Now, look again at the examples of improper fractions.

Let's try dividing the first one: . What is 4 ÷ 2? Yes, it's 2. So the improper fraction = 2. Now you try . Do you see that = 5? Notice the pattern here. Improper fractions are all equal to or greater than 1.

Mixed Numbers

Many improper fractions are equal to whole numbers. For example, = 2 and = 5. But some improper fractions are not equal to a whole number. They represent a whole number plus a proper fraction. A whole number plus a proper fraction is called a mixed number. Examples of mixed numbers are and so on.

When working with mixed numbers, it is usually easiest to change the mixed number to an improper fraction and then perform the given operations. To convert a mixed number to an improper fraction, multiply the whole number part by the denominator. Add this product to the numerator. This sum is the numerator of the improper fraction. The denominator stays the same.

Example

Convert to an improper fraction.

Multiply the whole number part by the denominator: 5 × 3 = 15.

Add this product to the numerator: 15 + 2 = 17.

The improper fraction is .

To convert an improper fraction to a mixed number, divide the numerator by the denominator. Find the whole number part, and the remainder becomes the numerator of the fractional part of the mixed number. The denominator stays the same.

Example

Convert to a mixed number. Find how many times 8 divides into 62:

Find the whole number part: 7.

Find the remainder: 6.

The mixed number is .

To perform mixed-number operations, convert to improper fractions.

Example

Change to improper fractions: .

Find the common denominator: .

Perform the subtraction: .

Change the improper fraction to a mixed number (if needed): .

Practice problems for these concepts can be found at:

Fractions and Decimals Practice Problems: GED Math