# You need to create a common denominator

### Question:

My son is learning fractions, and he has a pretty good handle on how to add and subtract them when the bottom number is the same. For example, he can add 5/7 and 1/7. But, we are having trouble understanding how to add fractions when the numbers are different. How do you add 3/8 and 1/4??

The solution, as your son's teacher has probably emphasized, is to find a common denominator. The denominator, by the way, is the bottom number.

The reason you need a common denominator is that you can't add fractions with different denominators. It just doesn't work that way, unfortunately. So, the only way to do it is to have the same number on the bottom of both fractions!

One way to find a common denominator is to multiply the two denominators together. For example, when adding 1/2 to 1/3, the common denominator is 3*2=6. Don't just change the bottoms to 6, though. You have to change the tops as well. Since you multiplied the 2 times 3 to get 6, you have to multiply the top by 3 as well. Likewise for the second fraction, you have to multiply the top and bottom by 2, as shown below:

$$\frac{1}{2}+\frac{1}{3} = ?$$ $$\frac{1}{2}*\frac{3}{3}+\frac{1}{3}*\frac{2}{2}=$$

Notice that we didn't change anything when we multiplied by 3/3 or 2/2. Since each one of those reduces to 1/1, it's really the same as multiplying by 1, which doesn't change anything. That's exactly our goal -- to rewrite a fraction in an equivalent form but with a new denominator! For example, we might want to re-write 1/2 as 2/4, or 3/5 as 9/15. The fraction will no longer be in simplest terms, but it will be in a form where we can do some addition (or subtraction).

$$\frac{1}{2}*\frac{3}{3}+\frac{1}{3}*\frac{2}{2}= \frac{3}{6}+\frac{2}{6} = \frac{5}{6}$$

All we did was change 1/2 into 3/6, and we also changed 1/3 into 2/6. Those are equivalent, so we are allowed to do that. Now that they have the same denominator, we can just add the numerators! Don't make the mistake of adding the denominators -- that stays the same! Also -- subtraction works the exact same way. Check out these examples below, and see if you can figure out how we found the common denominator and added the fractions.

$$\frac{1}{2}+\frac{1}{5} = \frac{1}{2}*\frac{5}{5}+\frac{1}{5}*\frac{2}{2}= \frac{5}{10}+\frac{2}{10} = \frac{7}{10}$$ $$\frac{2}{7}+\frac{1}{3} = \frac{2}{7}*\frac{3}{3}+\frac{1}{3}*\frac{7}{7}= \frac{6}{21}+\frac{7}{21} = \frac{13}{21}$$ $$\frac{4}{5}-\frac{1}{2} = \frac{4}{5}*\frac{2}{2}-\frac{1}{2}*\frac{5}{5}= \frac{8}{10}-\frac{5}{10} = \frac{3}{10}$$