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

Practice problems for these concepts can be found at:

Fractions and Decimals Practice Problems: GED Math

Operations with Fractions—Overview

To do well when working with fractions, it is necessary to understand some basic concepts. Here are some math rules for fractions using variables:

Adding and Subtracting Fractions

How would you add 2 hours and $5? You can't. You can add and subtract only like objects: You can add $2 to $5 or 2 hours to 5 hours. It's the same with fractions. To add and subtract fractions, you need like fractions. Like fractions are fractions that have the same denominator. If the denominators are already the same, add or subtract the numerators and keep the denominator. Then, simplify if needed.

Example

Subtract the numerators: (2 – 7) = –5.

Retain the denominator in your final answer: .

When subtracting fractions, the order of the fractions is important. Write the numerator that you are subtracting from first. Then subtract as you would any two numbers.

Fractions that have different denominators are called unlike fractions. Before you can add or subtract unlike fractions, you first need to change them into like fractions so that they have the same number in the denominator. This is called finding a common denominator.

There are two main ways to find a common denominator. One way is to multiply the denominators together. The other way is to multiply each denominator by 2, 3, 4, 5, and so on. Then compare the lists of multiples of each denominator. The numbers that are the same, or that are in common, are common denominators.

Follow these steps when adding or subtracting unlike fractions.

Step 1 Find a common denominator.

Step 2 Change each fraction so that it has the common denominator.

Step 3 Add or subtract the fractions as indicated.

Step 4 Reduce your answer to lowest terms.

Examples

Find a common denominator. The LCD of 4 and 24 is 24.

Convert the first fraction to have a denominator of 24: .

Perform the addition: .

Finally, simplify: .

Find a common denominator for and .

List the multiples for each denominator.

Multiples of 4: 4, 8, 12, 16,…

Multiples of 6: 6, 12, 18, 24,…

The numbers 4 and 6 share the multiple 12. So, 12 is a common denominator for and . In fact, it is the LCD.

Find a common denominator for and .

Multiply the denominators together: 4 × 6 = 24. So, 24 is a common denominator for and . However, it can be reduced to 12, which is the LCD.

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: