How do you subtract fractions?
Short answer. Rewrite both fractions over a common denominator, subtract the numerators, and keep the denominator: 3/4 − 1/6 = 9/12 − 2/12 = 7/12. With mixed numbers, if the first fraction part is too small, borrow 1 from the whole number: 4 1/5 − 1 2/3 = 3 18/15 − 1 10/15 = 2 8/15.
Subtracting fractions is adding fractions with the sign turned round. The machinery is identical. The only new part is borrowing, and borrowing is where the marks go.
Same denominator: one step
The denominator names the size of the pieces. The numerator counts them. Taking pieces away changes how many you have, not how big they are:
5/7 − 2/7 = 3/7 count the sevenths: 5 − 2 = 3 not 3/0. Subtract the denominators too and you have divided by zero, which is not an answer.
Different denominators: the full method
Take 3/4 − 1/6. Quarters and sixths are different sizes, so find a piece size both fit into. The smallest number 4 and 6 both divide into is 12, the least common denominator:
3/4 = 9/12 multiply top and bottom by 3 1/6 = 2/12 multiply top and bottom by 2 9/12 − 2/12 = 7/12
Subtraction has a free check that division does not: add the answer back. 7/12 + 2/12 = 9/12, which is 3/4, the number you started with. If the check does not land, something slipped.
The cross-multiply shortcut
If hunting for the least common denominator feels like work, multiply the denominators together and cross-multiply the tops. Keep the order, because subtraction cares which one comes first:
a/b − c/d = (a×d − c×b) / (b×d) 5/6 − 1/4 = (5×4 − 1×6) / 24 = (20 − 6) / 24 = 14/24 = 7/12
It always works, and the price is a bigger fraction to reduce at the end. Through the least common denominator the same sum is 10/12 − 3/12 = 7/12, with nothing left to simplify.
Mixed numbers, when nothing needs borrowing
If the first fraction part is at least as big as the second, subtract wholes from wholes and fractions from fractions:
5 3/4 − 2 1/3 Wholes 5 − 2 = 3 Fractions 3/4 − 1/3 = 9/12 − 4/12 = 5/12 Total 3 5/12
Mixed numbers, when you have to borrow
Now 4 1/5 − 1 2/3. Over fifteenths it reads 4 3/15 − 1 10/15, and 3/15 minus 10/15 goes below zero. So borrow, exactly as you would borrow a ten in column subtraction, except the thing you borrow is one whole, and one whole is 15/15:
4 1/5 − 1 2/3 = 4 3/15 − 1 10/15 Borrow 4 3/15 = 3 + 15/15 + 3/15 = 3 18/15 Fractions 18/15 − 10/15 = 8/15 Wholes 3 − 1 = 2 Total 2 8/15
If borrowing makes you nervous, skip it. Convert both to improper fractions and subtract those. It takes bigger numbers and no judgement:
4 1/5 = 21/5 = 63/15 1 2/3 = 5/3 = 25/15 63/15 − 25/15 = 38/15 = 2 8/15
Same answer. The borrowing route keeps the numbers small, the improper route never asks you to decide anything. Pick whichever one you are less likely to fumble. The multiplication and division of mixed numbers only work the improper way, so it is the habit worth building.
A whole number minus a fraction
Same borrowing, with nothing in the fraction column to start with. Write one of the wholes as a full set of pieces:
3 − 3/8 = 2 8/8 − 3/8 = 2 5/8
When the answer goes negative
Subtraction is not commutative. Swap the order and the sign flips. 1/2 − 1/3 is 1/6, and the other way round:
1/3 − 1/2 = 2/6 − 3/6 = −1/6
A negative fraction is a perfectly good answer. It just means the second amount was the bigger one.
Where this actually comes up
Kitchens and workshops, mostly. You have 3 1/8 cups of flour and the recipe wants 2 3/4, so you borrow: 2 9/8 − 2 6/8 leaves 3/8 of a cup. A cut of 2 5/8 inches off an 8 inch board leaves 5 3/8. A meeting booked for 2 1/4 hours that has already run 1 1/2 has 3/4 of an hour, 45 minutes, to go.
The decimal trap
Converting to decimals first is tempting, and subtraction is where it does the most damage. 1/3 has no exact decimal form, so any decimal you type is already rounded. Adding keeps that error small next to a big answer. Subtracting two close numbers leaves a small answer with the same error inside it:
1/3 − 1/4 as decimals: 0.33 − 0.25 = 0.08
exactly: 4/12 − 3/12 = 1/12 = 0.08333...
2/3 − 1/3 as decimals: 0.67 − 0.33 = 0.34
exactly: 1/3 = 0.3333...
The first is 4% low and the second is 2% high, from rounding that looked harmless. It is the same disease as 0.1 + 0.2 = 0.30000000000000004, where a calculator stores numbers in binary floating point instead of keeping them whole. Stay in fractions until the last line, and if you need a decimal at the end, convert the fraction to a decimal once, then.
Common mistakes, quickly
| Mistake | What happens | Fix |
|---|---|---|
| Subtracting denominators5/7 − 2/7 = 3/0 | Division by zero, which is not a number | Denominator stays put when piece sizes match |
| Subtracting straight across3/4 − 1/6 as (3 − 1)/(4 − 6) | 2/−2 = −1, negative when the answer is plainly positive | Common denominator first: 7/12 |
| Borrowing ten instead of one whole4 3/15 as 3 13/15 | 2 3/15 = 2 1/5 instead of 2 8/15 | The borrowed 1 is 15/15, not 10 |
| Swapping fractions to dodge borrowing4 1/5 − 1 2/3 as 3 + (2/3 − 1/5) | 3 7/15 instead of 2 8/15 | Borrow, or convert to improper fractions |
| Rounding to decimals first0.33 − 0.25 | 0.08, which is 4% below 1/12 | Stay in fractions until the end |
Every number on this page was computed exactly and checked before publishing.
Or type 4 1/5 − 1 2/3 and get 2 8/15 back
Prism has a fraction mode that subtracts exactly, takes mixed numbers as typed and reduces for you, so 1/3 − 1/4 is 1/12 and not 0.08. Fractions are free forever, no subscription. See the app.