Subtracting fractions
Subtracting fractions works just like adding: find a common denominator, subtract the numerators, simplify. Mixed numbers add one wrinkle — borrowing — covered below.
Working
- Find the least common denominator of 6 and 4: LCD = 12.
- Rewrite each fraction over 12.
5/6 = 10/12, 1/4 = 3/12 - Subtract the numerators and keep the denominator.
10 − 3 = 7 → 7/12 - 7/12 is already in lowest terms.
How to subtract fractions, step by step
- Make the denominators match. Find the LCD. For 5/6 − 1/4 it is 12.
- Rewrite both fractions. 5/6 = 10/12 and 1/4 = 3/12.
- Subtract the numerators. 10/12 − 3/12 = 7/12. The denominator stays the same.
- Simplify. 7/12 is already in lowest terms. If the result is negative, the second fraction was bigger.
Why fractions need a common denominator
The denominator names the size of the pieces. You can only subtract pieces of the same size — so first we re-cut the bars. Step through it:
Step 1 of 3 — Here are 1/2 and 1/3. The pieces are different sizes, so "1 piece minus 1 piece" means nothing yet.
Subtracting mixed numbers with borrowing
Take 3 1/4 − 1 3/4. You cannot take 3/4 from 1/4, so borrow 1 from the 3: 3 1/4 = 2 5/4. Now 2 5/4 − 1 3/4 = 1 2/4 = 1 1/2.
The alternative is to convert both to improper fractions: 13/4 − 7/4 = 6/4 = 1 1/2. No borrowing needed, which is why the calculator works this way.
Subtracting a fraction from a whole number
Borrow one whole as a fraction: 5 − 2/3 = 4 3/3 − 2/3 = 4 1/3.
When the answer is negative
1/3 − 1/2 = 2/6 − 3/6 = −1/6. A negative answer is correct whenever the second fraction is larger than the first — compare them first if you are not sure.
Common mistakes
- Subtracting the denominators: 3/4 − 1/2 ≠ 2/2.
- Subtracting the smaller fraction part from the larger in a mixed number instead of borrowing: 3 1/4 − 1 3/4 is not 2 2/4.
- Forgetting to reduce the whole number after borrowing.
Questions people ask
How do you subtract fractions with different denominators?
Rewrite both over their least common denominator, subtract the numerators and keep the denominator, then simplify. 3/4 − 1/6 = 9/12 − 2/12 = 7/12.
What is 1 − 1/3?
Write 1 as 3/3: 3/3 − 1/3 = 2/3.
What is 3/4 − 1/2?
1/2 = 2/4, so 3/4 − 2/4 = 1/4.
How do you borrow when subtracting mixed numbers?
Take 1 from the whole number and add it to the fraction as denominator/denominator. 4 1/5 becomes 3 6/5, and then you can subtract the fraction parts normally.