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Adding and Subtracting Fractions Study Guide

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Updated on Oct 4, 2011

Introduction to Adding and Subtracting Fractions

I know that two and two make four—and should be glad to prove it too if I could—though I must say if by any sort of process I could convert two and two into five it would give me much greater pleasure.

—George Gordon, Lord Byron, British poet (1788–1824)

In this lesson, you will learn how to add and subtract fractions and mixed numbers.

Adding and subtracting fractions can be tricky. You can't just add or subtract the numerators and denominators. Instead, you have to make sure that the fractions you're adding or subtracting have the same denominator before you do the addition or subtraction.

Adding Fractions

If you have to add two fractions that have the same bottom numbers, just add the top numbers together and write the total over the bottom number.

Example: , which can be reduced to Note: There are a lot of sample questions in this lesson. Make sure you do the sample questions and check your solutions against the step-by-step solutions at the end of this lesson before you go on to the next section.

Finding the Least Common Denominator

To add fractions with different bottom numbers, raise some or all the fractions to higher terms so they all have the same bottom number, called the common denominator. Then add the numerators, keeping the denominators the same.

All the original bottom numbers divide evenly into the common denominator. If it is the smallest number that they all divide evenly into, it is called the least common denominator (LCD). Addition is often faster using the LCD than it is with just any old common denominator.

Here are some tips for finding the LCD:

  • See if all the bottom numbers divide evenly into the largest bottom number.
  • Check out the multiplication table of the largest bottom number until you find a number that all the other bottom numbers divide into evenly.

Tip

The fastest way to find a common denominator is to multiply the two denominators together. Example: For and you can use 4 × 8 = 32 as your common denominator.

Example: 

1. Find the LCD by multiplying the bottom numbers: 3 × 5=15
2. Raise each fraction to 15ths, the LCD:
   
3. Add as usual:

Adding Mixed Numbers

Mixed numbers, you remember, consist of a whole number and a fraction together. To add mixed numbers:

  1. Add the fractional parts of the mixed numbers. (If they have different bottom numbers, first raise them to higher terms so they all have the same bottom number.)
  2. If the sum is an improper fraction, change it to a mixed number.
  3. Add the whole number parts of the original mixed numbers.
  4. Add the results of steps 2 and 3.

Example:  

1. Add the fractional parts of the mixed numbers:
2. Change the improper fraction into a mixed number:
3. Add the whole number parts of the original mixed numbers:
4. Add the results of steps 2 and 3:

Subtracting Fractions

As with addition, if the fractions you're subtracting have the same bottom numbers, just subtract the second top number from the first top number and write the difference over the bottom number.

Example:  

To subtract fractions with different bottom numbers, raise some or all of the fractions to higher terms so they all have the same bottom number, or common denominator, and then subtract. As with addition, subtraction is often faster if you use the LCD rather than a larger common denominator.

Example:  

  1. Find the LCD. The smallest number that both bottom numbers divide into evenly is 12. The easiest way to find it is to check the multiplication table for 6, the larger of the two bottom numbers.
  2. Raise each fraction to 12ths, the LCD:

  3. Subtract as usual:

Subtracting Mixed Numbers

To subtract mixed numbers:

  1. If the second fraction is smaller than the first fraction, subtract it from the first fraction. Otherwise, you'll have to "borrow" (explained by example further on) before subtracting fractions.
  2. Subtract the second whole number from the first whole number.
  3. Add the results of steps 1 and 2.

Example

1. Subtract the fractions:
2. Subtract the whole numbers:
3. Add the results of steps 1 and 2:

When the second fraction is bigger than the first fraction, you'll have to perform an extra "borrowing" step before subtracting the fractions.

Example

1. You can't subtract the fractions the way they are because is bigger than . So you have to "borrow":
  • Rewrite the 7 part of as :
(Note: Fifths are used because 5 is the bottom number in ; also, .)
  • Then add back the part of :
2. Now you have a different version of the original problem:
3. Subtract the fractional parts of the two mixed numbers:
4. Subtract the whole number parts of the two mixed numbers:
5. Add the results of the last 2 steps together:

Tip

Don't like the borrowing method previously shown? Here's another way to subtract mixed fractions:

  • Change mixed fractions to improper fractions.

  • Find common denominators.

  • Subtract fractions: subtract the numerators and keep the denominator the same.

  • If the answer is an improper fraction, change it back into a mixed number.

 

Tip

The next time you and a friend decide to pool your money together to purchase something, figure out what fraction of the whole each of you will donate. Will the cost be split evenly: for your friend to pay and for you to pay? Or is your friend richer than you and offering to pay of the amount? Does the sum of the fractions add up to one? Can you afford to buy the item if your fractions don't add up to one?

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