To add and subtract fractions with different denominators, you must convert the fractions into fractions that have the same denominator with the appropriate numerator. The steps for adding and subtracting fractions are very similar to the last step, when you have to add and subtract the numerator of the fractions. If you want to know how to add and subtract fractions with different denominators, just follow these steps.
Step
Method 1 of 2: Finding Common Denominators
Step 1. Place the fractions next to each other
Write down the fractions you are working with next to each other. Put the numerator (top number) on the same level as the other numerators above, and the denominator (bottom number) in line below it. Let's use the fractions 9/11 and 2/4 as our examples.
Step 2. Understand equivalent fractions
If you multiply the numerator and denominator of a fraction by the same number, you get an equivalent fraction, just like the original fraction. For example, if you take 2/4, and multiply each number by 2, you get 4/8, which is the same ("equivalent") fraction as 2/4. You can check this for yourself by describing the fraction:
- Draw a circle, divide it into four equal parts, then color two of the four parts (2 / 4).
- Draw a new circle, divide it into 8 equal parts, then color four of the 8 parts (4/8).
- Compare the colored areas of the two circles, representing 2/4 and 4/8. Both are the same size.
Step 3. Multiply two denominators to find a common denominator
Before we can add or subtract fractions, we must write them down so that the fractions have the same denominator that is divisible by both of the denominators. The quickest way to find it is to multiply the two denominators. Once you've written down the answer, you can move on to the solving part of the problem, or try the steps below to find the same denominator but in a different way, which may be easier to work with.
- For example, let's start with the fractions 9/11 and 2/4. 11 and 4 are the denominators.
- Multiply both denominators: 11 x 4 = 44.
Step 4. Find the same smaller denominator (optional)
The above method is fast, but you can search for "smallest common denominator", meaning the smallest possible answer. To do this, write down a multiple of each initial denominator. Circle the smallest number that appears on both lists of multiples. Here's a new example, which we can use if we solve "5/6 + 2/9":
- The denominators are 6 and 9, so we have to "count six-six" and "count nine-nine" to find multiples:
-
Multiple of
Step 6.: 6, 12
Step 18., 24
-
Multiple of
Step 9.: 9
Step 18., 27, 36
-
Because
Step 18. are in both tables, 18 can be used as the common denominator.
Method 2 of 2: Solving Problems
Step 1. Change the first fraction to use the same denominator
In our first example, using 9/11 and 2/4, we decided to use 44 as the common denominator. But remember, you can't just change the denominator without multiplying the numerator by the same number. Here's how we convert fractions to equivalent fractions:
-
We know that 11 x
Step 4. = 44 (this is how we get 44, but you can also solve 44 11 if you forgot).
- Multiply both sides of the fraction by the same number to get the result:
-
(9 x
Step 4.) / (11
Step 4.) = 36/44
Step 2. Do the same for the second fraction
Here's the second fraction in our example, 2/4, converted to a fraction equivalent to 44 as the denominator:
-
4 x
Step 11. = 44
-
(2 x
Step 11.) / (4
Step 11.) = 22/44.
Step 3. Add or subtract the numerators of the fractions to get the answer
After both fractions share 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
Step 4. Convert common fractions to mixed numbers
If the numerator is greater than the denominator, you have a fraction greater than 1 (a "regular" fraction). You can convert it to a mixed number, which is easier to read, by dividing the numerator by the denominator, and putting the remainder as a fraction. For example, using the fraction 58 / 44, we get 58 44 = 1, with a remainder of 14. This means that our final mixed number is 1 and 14/44.
- If you're not sure how to divide the number, you can continue to subtract the bottom number from the top number, writing down the number of times you've subtracted. For example, change 317/100 like this:
-
317 - 100 = 217 (subtract
Step 1. time). 217 - 100 = 117 (subtract
Step 2. time). 117 - 100 = 17
Step 3. time). We can't subtract any more, so the answer is 3 and 17/100.
Step 5. Simplify the fraction
Simplifying a fraction means writing it in its least equivalent form, to make it easier to use. Do this by dividing the fraction and denominator by the same number. If you can find a way to simplify the answer again, keep doing it until you don't find it. For example, to simplify 14/44:
- The numbers 14 and 44 are divisible by 2, so let's use them.
- (14 ÷ 2) / (44 ÷ 2) = 7 / 22
- No other number is divisible by 7 and 22, so here's our simplified final answer.
Sample Questions
Try solving these problems yourself. If you think you already know the answer, block or select the invisible text after the equals sign, to read the answer and check your work. The questions in each section will get harder as you go down. The last questions are tricky, so don't expect to find the answer 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
