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Subtracting Fractions with Borrowing   
by Gail Englert

A response to the question:

How do you subtract a fraction with whole numbers when you have to borrow?

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How do you subtract a fraction with whole numbers when you have to borrow? For example: What is 99 1/5 - 66 4/10?


Start out with an estimate. 99 1/5 is really just a little bit bigger than 99, and 66 4/10 is almost 66 1/2, so now you have the problem 99 - 66 1/2. 99 - 66 is 33, so 99 - 66 1/2 must be 32 1/2.

Now that you have a reasonable (ballpark) answer for your problem, you can work on the actual problem. I ask my fifth grade students to think about several different ways they can represent fifths. Some people call that "finding equivalent fractions". My students know that 1/5 can also be called 2/10, 3/15, 4/20, 5/25, 6/30... (If you look at all those fractions, you will see there is a pattern).

Now let's look at 4/10. It can also be called 2/5, (4/10), 6/15, 8/20, 10/25, 12/30... Do you see the pattern this time?

So, if we compare the two sets of fractions, we can see there are several different ways to match the two fractions up so that they have the same denominator (bottom number). We could use 1/5 and 2/5, 2/10 and 4/10, 3/15 and 6/15, etc. The pair you choose won't really make any difference, but in the end, if you want to do the least work possible, you are probably best off choosing the pair with the smallest denominators, which, in this case, would be 1/5 and 2/5.

So now you have the problem 99 1/5 - 66 2/5. What would happen if you could take one of the wholes away from 99, and change it into fifths? You would have one less whole, and 5 more fifths. Instead of 99 1/5 you would have 98 6/5. Now your problem is 98 6/5 - 66 2/5. That is an easier problem to solve.

Don't be discouraged by the number of steps this seems to take. The more of them you do, the more you will remember about equivalent fractions, and you will be able to begin to find shortcuts. Think of finding your way through a building or a new city you are not familiar with. At first, you have to think about every step you take, and consult a map or directions often. Then, after a while, you are able to recognize landmarks, and soon you are navigating without any trouble at all. That is the way it is with mathematics. At first it all seems strange, but after practice, you can do it!

-Gail, for the T2T service

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