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Fraction Addition and Subtraction

  /  Fraction Addition and Subtraction

Sharon Vogt

Adding or subtracting fractions can be quite a challenge. Doing one problem requires several steps. Each step has its own set of skills. But, no need to worry. In this article you’ll learn all the steps you need to work with fractions. 

Below is a list of the skills needed to solve addition and subtraction problems. If you need a brush up on one of these skills, just click on the link. You’ll be taken to the part of the problem that deals with that particular skill.

  • Least Common Denominator (LCD) 
  • Also known as Least Common Multiple (LCM)
  • Equivalent Fractions
  • Fraction Addition
  • Reducing Fractions to Simplest Form
  • Renaming Fractions
  • Mixed Number Subtraction

Least Common Denominator

Before adding or subtracting fractions, you have to have denominators that are the same. You can’t add 1/4 and 1/3 because the sections are not the same size. It?s like the old saying, you can?t add apples and bananas.

 

1/4
1/3

 

To add these two fractions, you need a common denominator. 

The denominator is the bottom number of a fraction. A common denominator simply means that they are both the same. 

To find a common denominator, compare the multiples of the numbers in the denominators. Remember, the common denominator is the same as a common multiple.

But what are multiples? Multiples are the answers to multiplication problems.

These multiplication problems give you the multiples of 4.

1 x 4 = 4

2 x 4 = 8

3 x 4 = 12

4 x 4 = 16

4 x 5 = 20

4 x 6 = 24

4 x 7 = 28

4 x 8 = 32

The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32 and so on.

These multiplication problems give you the multiples of 3.

1 x 3 = 3

2 x 3 = 6

3 x 3 = 9

3 x 4 = 12

3 x 5 = 15

3 x 6 = 18

3 x 7 = 21

3 x 8 = 24

The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24 and so on . . .

Inthese lists of multiples, you can find common multiples. Both 12 and 24 appear in both lists of multiples. 12 and 24 are common multiples of 3 and 4. 

The least common multiple is the multiple that has the least value. 12 is less than 24, so 12 is the least common multiple.

12 is also the least common denominator.

Quick Tip:  Sometimes when working with fractions, the hardest thing to find is a common denominator. This doesn?t have to be the LEAST common multiple to work. It just has to be a common multiple. To find one quickly, just multiply the denominators.

For the fractions 2/8 and 1/6, you could use  8 x 6, or 48 as the common denominator, even though the least common denominator is 24.

Equivalent Fractions

After finding a common denominator, the next step is to write equivalent fractions. This just means that you will write new fractions with the new denominators.

Let’s go back to the fractions 1/4 and 1/3 and write them with the least common denominator – 12.

For each fraction, ask yourself, what do you have to do to the original fraction to make it have the new denominator.

1/4 = ?/12

What do you have to do to the 4 to make it 12? You have to multiply 4 by 3. Do the same to the numerator. Remember, the numerator is the top number of a fraction.

1 x 3 = 3
4 x 3 = 12

1/4 = 3/12

Do the same for the second fraction.

1/3 = ?/12

What do you have to do to 3 to change it to 12. That’s right, multiply by 4. But remember to also multiply the numerator by 4.

1 x 4 = 4
3 x 4 = 12

1/3 = 4/12

Equivalent fractions for 1/4 and 1/3 are 3/12 and 4/12.

Fraction Addition

Now that you have fractions with common denominators, adding fractions is simple. Just add the numerators, or top numbers. Write the sum over the denominator, or bottom number.

3/12 + 4/12 = 7/12

So, 1/4 + 1/3 = 7/12

Reducing Fractions to Simplest Form

Reducing fractions makes fractions easier to understand. It may be hard to picture what 25/75 might look like. But, if you reduce this fraction, you?ll find out it?s a fraction you are very familiar with.

Look at the numbers in the fraction 25/75. What number could you divide BOTH numbers by without a remainder? Since they both end in 5, they are both multiples of 5. (This doesn?t work for all numbers, but does work for 5s.) That means you can divide both numbers evenly by 5.

25 ÷ 5 = 5
75 ÷ 5 = 15

25/75 reduces to 5/15. 

Can 5/15 be reduced? Is there a number that will evenly divide both 5 and 15?
Yes, 5 will divide both numbers evenly. So, divide again.

5 ÷  5 = 1
15 ÷ 5 = 3

5/15 reduces to 1/3. 

Can 1/3 be reduced? Is there a number that will evenly divide both 1 and 3?
Yes, 1 divides both 1 and 3. But dividing by 1 will not change the numbers in the fraction. 

Since the only number left to divide by is 1, this fraction is said to be in simplest form. 

25/75 in simplest form is 1/3.

The addition problem we solved above had a sum of 7/12. Is 7/12 in simplest form?

The only numbers that divide 7 are 1 and 7. 7 will not divide 12 evenly. So, the only number that will divide 7 and 12 is 1.

7/12 is in simplest form.

Renaming Fractions

Sometimes when you subtract mixed numbers, you need to be able to borrow, or rename.
Here’s an example.

4  3/8
-1  7/8

You cannot take 7/8 from 3/8. Since there aren?t enough eighths to subtract from, we need to create more eighths. You can do this by taking 1 from the 4 and changing it to eighths.

Remember, the whole number 1 can be written as a fraction by writing any number over itself.

1 = 4/4 or 8/8 or 17/17 or 235/235

This is true because any number divided by itself is 1. The fraction bar is simply a division sign.

4 ÷ 4 = 1 and 8  ÷ 8 = 1 and 235 ÷ 235 = 1

So, to make more eighths, take 1 from the 4 in 4  3/8 and change it to 8/8.

4 3/8 =
3 + 1 + 3/8 =
3 + 8/8 + 3/8 =
3 + 11/8

Rename 4 3/8 as  3 11/8.

Quick Tip:  There?s a short cut for renaming fractions. Just subtract 1 from the whole number. Add the numerator and denominator and write the sum over the denominator.

7 3/5 = 6 8/5

Mixed Number Subtraction

In the previous section, we used the following problem as an example.

4  3/8
– 1  7/8

Let’s use what you know about renaming fractions to solve this problem.

4  3/8 = 3  11/8
-1  7/8 = 1  7/8

Now that the first fraction has been renamed, just subtract the numerators of the fractions.

11/8 – 7/8 = 4/8

Subtract the whole numbers, too.

3 – 1 = 2

4  3/8 = 3  11/8
-1  7/8 = 1  7/8

2 4/8

Before you can say you are done with this problem, you must check to see if the fraction part is in simplest form.

Is there any number you can divide both 4 and 8 by without getting a remainder?
You can divide 4 and 8 by either 2 or 4. Choose the greater number. Then you won?t have to divide again.

4 ÷ 4 = 1
8 ÷ 4 = 2

4/8 reduces to 1/2.

2 4/8 = 2 1/2

So, 4  3/8  – 1  7/8 = 2  1/2.

Copyright © Sharon Vogt is a freelance author of math educational materials. She has written for several major textbook publishers including MacMillan/McGraw-Hill, ScottForesman, and Addison Wesley. Ms. Vogt has also written several teacher/student resource books published by Frank
Schafer Publications, GoodYear Books, and Carson Dellosa. You may read more about her books currently in print at Barnes and Noble Prior to her writing career, Sharon taught math to students in grades 5 through 9.