Add and Subtract Fractions With Unlike Denominators 

In order to add and subtract fractions with unlike denominators, you have to convert them into fractions with like denominators and corresponding numerators. The steps for adding and subtracting fractions are very similar until the very end, when you have to either add or subtract the numerators of the fractions. If you want to know how to add and subtract fractions with unlike denominators, just follow these steps.

Steps

Finding a Common Denominator

  1. Place the fractions side by side. Write the fractions you are using side by side. Keep the numerators (top numbers) level with each other on top, and the denominators (bottom numbers) on the line beneath them. Let's use the fractions 9 / 11 and 2 / 4 as our example.
  2. Understand equivalent fractions. If you multiply the numerator and denominator in a fraction by the same number, you end up with an equivalent fraction, exactly equal to the first. For example, if you take 2 / 4, and multiply each number by 2, you get 4 / 8, which is an equal ("equivalent") fraction to 2/4. You can test this yourself by drawing the fractions:
    • Draw a circle, divide it into four equally sized pieces, then color in two of them (2 / 4).
    • Draw a new circle, divide it into 8 equal pieces, then color in four of them (4 / 8).
    • Compare the colored areas in the two circles, representing 2/4 and 4/8. These two areas have an equal size.
  3. Multiply the two denominators to find a common denominator. Before we can add or subtract the fractions, we need to write them so they have the same denominator (a "common denominator") that is divisible by both numbers. The quickest way to find this is to multiply the two denominators together. Once you've written the answer down, you can skip ahead to the section on Add-and-Subtract-Fractions-With-Unlike-Denominators, or try the step below to find a different common denominator that may be easier to use.
    • For example, we started with the fractions 9 / 11 and 2 / 4. 11 and 4 are the denominators.
    • Multiply the two denominators together: 11 x 4 = 44.
  4. Find a smaller common denominator instead (optional). The method above is quick, but you can instead find the "least common denominator," meaning the smallest possible answer. To do this, write down the multiples for each of the original denominators. Circle the smallest number that appears on both lists. Here's a new example, which we could use if we were solving "5/6 + 2/9":
    • The denominators are 6 and 9, so we want to "count by sixes" and "count by nines" to find the multiples:
    • Multiples of 6: 6, 12, 18, 24
    • Multiples of 9: 9, 18, 27, 36
    • Since 18 is in both tables, it can be used as a common denominator.

Finishing the Problem

  1. Change the first fraction to use the common denominator. In our first example, using 9/11 and 2/4, we decided to use 44 as the common denominator. But remember, we can't just change the denominator without multiplying the numerator by the same amount as well. Here's how we turn it into an equivalent fraction:
    • We know 11 x 4 = 44 (this is how we found the number 44 to begin with, but you can solve 44 ÷ 11 if you forgot).
    • Multiply both sides of the fraction by the same number to get the result:
    • (9 x 4 ) / (11 x 4) = 36/44
  2. Do the same for the second fraction. Here's the second fraction in our example, 2 / 4, transformed into an equivalent fraction using 44 as the denominator:
    • 4 x 11 = 44
    • (2 x 11) / (4 x 11) = 22/44.
  3. Add or subtract the numerators of the fractions to get the answer. Once both fractions use the same denominator, you can add or subtract the numerators to get the answer:
    • Addition: 36 / 44 + 22 / 44 = (36 + 22) / 44 = 58/44
    • Or subtraction: 36 / 44 - 22/44 = (36 - 22) / 44 = 14 / 44
  4. Convert improper fractions into a mixed number. If the numerator ends up larger than the denominator, you have a fraction larger than 1 (an "improper fraction). You can make these into a mixed number, which is easier to read, by dividing the numerator by the denominator, and keeping the remainder as a fraction. For example, using the fraction 58 / 44, we get 58 ÷ 44 = 1, with remainder 14 left over. This means our final mixed number is 1 and 14/44.
    • If you're not sure how to divide the numbers, you can keep subtracting the bottom number from the top, writing down how many times you've subtracted. For example, convert 317 / 100 like this:
    • 317 - 100 = 217 (subtracted 1 time). 217 - 100 = 117 (subtracted 2 times). 117 - 100 = 17 (3 times). We can't subtract any more, so the answer is 3 and 17/100.
  5. Simplify the fraction. Simplifying a fraction means writing it in its smallest equivalent form, to make it easier to use. Do this by dividing the numerator and denominator by the same number. If you can find a way to simplify the answer even further, keep doing it until you can't find another. For example, to simplify 14/44:
    • The numbers 14 and 44 are both divisible by 2, so let's use that.
    • (14 ÷ 2 ) / (44 ÷ 2) = 7 / 22
    • There are no numbers that divide evenly into both 7 and 22, so this is our final, simplified answer.

Example Problems

  • Try to solve these problems on your own. When you think you have the answer, Select-Phrases-of-Text-on-an-iOS-Device the invisible text after the equal sign to read the answer and check your work. The problems in each section get harder as you move down the list. The last ones are tricky, so don't expect to get every one on the first try:

Practice addition problems:

  • 1 / 2 + 3 / 8 = 7 / 8
  • 2 / 5 + 1 / 3 = 11 / 15
  • 3 / 4 + 4 / 8 = 1 and 1/4
  • 10 / 3 + 3 / 9 = 3 and 2/3
  • 5 / 6 + 8 / 5 = 2 and 13/30
  • 2 / 17 + 4 / 5 = 78 / 85

Practice subtraction problems:

  • 2 / 3 - 5 / 9 = 1 / 9
  • 15 / 20 - 3 / 5 = 3 / 20
  • 7 / 8 - 7 / 9 = 7 / 72
  • 3 / 5 - 4 / 7 = 1 / 35
  • 7 / 12 - 3 / 8 = 5 / 24
  • 16 / 5 - 1 / 4 = 2 and 19/20



Tips

  • The Least Common Denominator is shortened as "LCD." "Least" means "smallest" and "common" means "shared", so in more modern language you can think of it as the "smallest shared denominator" you can use for both fractions.

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