How can I subtract fractions without subtracting the denominators?
Quick answer
First rewrite the fractions so they have the same denominator. That makes both fractions count the same-sized parts. Then subtract the numerators and keep the shared denominator; simplify if possible. For example, 5/6 − 1/4 becomes 10/12 − 3/12 = 7/12. Check that the answer is smaller than 5/6, since you took away a positive amount.
The question, the details, the next step.
Why can’t I just subtract the denominators too? If I have 5/6 − 1/4, why isn’t the answer 4/2?
Because the denominators tell you the size of the pieces, not how many pieces you have. Sixths and fourths are different-sized pieces, so subtracting the numerators while keeping the original denominators would mix unlike units. It is like trying to subtract three small slices from ten larger slices without first expressing them in the same-sized pieces.
For subtraction, first give both fractions a common denominator. Then the numerators count the same kind of piece, so you can subtract those counts. The method is to use a common denominator, subtract the numerators, keep the denominator, and simplify the result if possible. [S1]
Can you show exactly how that works for 5/6 − 1/4?
One common denominator for 6 and 4 is 12. Rewrite each fraction as an equivalent fraction with denominator 12:
- 5/6 = 10/12, because both numerator and denominator are multiplied by 2.
- 1/4 = 3/12, because both are multiplied by 3.
- Subtract the numerators and keep the shared denominator: 10/12 − 3/12 = 7/12.
So the answer is 7/12. Both amounts are now measured in twelfths, which makes the subtraction straightforward. The result cannot be simplified further: 7 and 12 have no common factor greater than 1.
How do I check whether 7/12 is a sensible answer, not just a calculation?
Use the size of the amounts as a reasonableness check. You started with 5/6 and took away the positive amount 1/4, so the result should be smaller than 5/6. Since 5/6 is 10/12, comparing 7/12 with 10/12 shows that it is smaller. It is also still positive: you took away less than you started with. Both checks fit the situation.
This check is about the direction and rough size of the answer, not a replacement for the common-denominator calculation. In general, subtracting a positive amount should leave less than you began with; if the amount taken away is larger, the result can be negative. [S1]
What should I write down so I don’t make the denominator mistake on the next problem?
Use this copyable pattern and fill in each line:
[fraction] − [fraction]
Rewrite both as equivalent fractions with denominator [common denominator].
[numerator]/[common denominator] − [numerator]/[common denominator] = [difference]/[common denominator]
Simplify if possible.
Check: after subtracting a positive amount, is the result smaller than the starting fraction?
For this problem, the filled-in work is:
5/6 − 1/4
= 10/12 − 3/12
= 7/12
Check: 7/12 < 10/12 (= 5/6), so the result is smaller than the starting amount.
For different denominators, finding a least common denominator is a useful way to choose a shared denominator. [S1]
A practical checklist
Identify the denominators and choose a shared denominator.
Rewrite each fraction as an equivalent fraction with that denominator.
Subtract the numerators; keep the shared denominator.
Simplify the resulting fraction if possible.
Check that subtracting a positive amount left less than you started with.
A few more useful answers
Does the shared denominator have to be the least common denominator?
The least common denominator is a useful choice: it gives you a common denominator for rewriting the fractions before subtracting. [S1]
What if the amount I subtract is bigger than the starting fraction?
The result can be negative. For a reasonableness check, consider whether the amount taken away exceeds the starting amount; if it does, a negative answer may make sense. [S1]
Sources
References used for this guide.
- [S1] OpenStax Elementary Algebra: Add and Subtract Fractions openstax.org · Checked