Adding and Subtracting Fractions With Unlike Denominators

To add or subtract fractions with different bottoms, first rewrite them so both have the same denominator, then add or subtract only the numerators. The denominator stays the same. The Common Core Grade 5 standard describes exactly this: replace the given fractions with equivalent fractions that share a denominator, so the sum or difference is unchanged. Its example is 2/3 + 5/4 = 8/12 + 15/12 = 23/12, and the general form is a/b + c/d = (ad + bc)/bd.

1. Why the bottoms must match

The denominator names the size of the pieces. Thirds and fourths are different sizes, so you cannot count them together any more than you can add 2 dollars and 3 quarters by writing "5 coins." Rewriting both fractions in twelfths turns them into the same kind of piece, and then counting works.

The Grade 4 standards lay the groundwork: a/b is equivalent to (n × a)/(n × b), and two fractions with different denominators can be compared by creating common denominators or by comparing to a benchmark such as 1/2. That multiplying-top-and-bottom move is the engine behind everything below.

2. The four steps

1. Find a common denominator, ideally the smallest one (the least common multiple of the two denominators).
2. Rewrite each fraction with that denominator by multiplying the top and bottom by the same number.
3. Add or subtract the numerators. Keep the denominator.
4. Simplify, and convert to a mixed number if the answer is more than 1.

3. Example 1: 1/2 + 1/3

The smallest number that both 2 and 3 divide into is 6.

1/2 = 3/6 and 1/3 = 2/6. Then 3/6 + 2/6 = 5/6.

Check with decimals: 0.5 + 0.333... = 0.833..., and 5/6 = 0.833.... It matches.

4. Example 2: 3/4 - 2/5

The common denominator is 20.

3/4 = 15/20 and 2/5 = 8/20. Then 15/20 - 8/20 = 7/20.

Check: 0.75 - 0.40 = 0.35, and 7/20 = 0.35.

5. Example 3: 5/12 + 7/18

List multiples to find the least common denominator: 12, 24, 36 and 18, 36. The smallest shared one is 36, not 12 times 18 = 216. Using the smallest keeps the numbers small, though any common denominator gives a correct answer after simplifying.

5/12 = 15/36 and 7/18 = 14/36. Then 15/36 + 14/36 = 29/36.

Does 29/36 simplify? 29 is prime and does not divide 36, so no. Check: 0.4167 + 0.3889 = 0.8056, and 29/36 = 0.8056.

6. Example 4: mixed numbers, 2 1/3 + 1 3/4

One way is to turn each mixed number into an improper fraction. 2 1/3 = 7/3 and 1 3/4 = 7/4. The common denominator is 12, so 7/3 = 28/12 and 7/4 = 21/12.

28/12 + 21/12 = 49/12. Since 49 divided by 12 is 4 with remainder 1, the answer is 4 1/12.

Another way is to add the whole numbers and fractions separately: 2 + 1 = 3, and 1/3 + 3/4 = 4/12 + 9/12 = 13/12 = 1 1/12. Then 3 + 1 1/12 = 4 1/12. Both routes agree.

7. Example 5: subtracting when you must regroup, 5 1/6 - 2 3/4

Convert: 5 1/6 = 31/6 and 2 3/4 = 11/4. The common denominator is 12, so 31/6 = 62/12 and 11/4 = 33/12.

62/12 - 33/12 = 29/12 = 2 5/12.

Check by estimating: 5.17 - 2.75 = 2.42, and 2 5/12 = 2.417. Close.

8. A word problem

A recipe uses 3/8 cup of oil and 2/3 cup of water. How much liquid is that?

The common denominator of 8 and 3 is 24. 3/8 = 9/24 and 2/3 = 16/24. The total is 25/24 cups, which is 1 1/24 cups. The answer is a little more than 1 cup, which makes sense because 3/8 is a bit under half and 2/3 is a bit over half.

9. The mistake the standard warns about

The most common error is adding tops and bottoms: 2/5 + 1/2 = 3/7. The Common Core standard uses this exact case as its example of an unreasonable answer. You can see it is wrong without any calculation: 3/7 is less than 1/2, yet you added something to 1/2, so the result must be larger than 1/2. The correct sum is 4/10 + 5/10 = 9/10.

Other slips to watch for:

• Changing the denominator on only one fraction, or changing the top without changing the bottom by the same factor.
• Adding the denominators in the final answer. The denominator stays put.
• Forgetting to simplify, or to convert a top-heavy answer such as 25/24 to a mixed number if the teacher asks for it.
• Subtracting the smaller fraction from the larger one in mixed numbers. In 5 1/6 - 2 3/4, the fraction 1/6 is smaller than 3/4, so you cannot subtract the fraction parts directly; regroup or use improper fractions as in Example 5.

10. A quick estimate before you start

Compare each fraction with the benchmarks 0, 1/2, and 1. For 3/4 - 2/5, 3/4 sits between 1/2 and 1, and 2/5 is just under 1/2, so the answer should be a bit more than 1/4. The result 7/20 = 0.35 fits. If your answer is far from the estimate, recheck the denominators first.

11. Key takeaways

• Denominators must match before you add or subtract.
• Multiply the top and bottom of each fraction by the same number to rewrite it.
• Add or subtract the numerators only; the denominator does not change.
• Simplify, and convert improper fractions when asked.
• Check by converting to decimals or by comparing with an estimate, as the Grade 5 standard recommends with benchmark fractions.