Arithmetic & Numbers · Grades 4–7
How to Add Fractions (Same and Different Denominators)
Why the denominators have to match
A denominator tells you the size of each piece. You can only count pieces together when they are the same size — three quarters plus two quarters is five quarters, but a half plus a third is not “two of anything” until you cut both into equal-sized slices.
That is why you never add the denominators. 1/2 + 1/3 is not 2/5: the 5 would describe a new piece size that you have not actually created.
Same denominators: the easy case
When the denominators already match, add the numerators and keep the denominator unchanged. 2/9 + 4/9 is 6/9 — and then you simplify, because 6 and 9 share a factor of 3.
Same denominator, then simplify
- Decimal: 0.666667
Step-by-step working 6 steps
- 1Write the problem
2/9 + 4/9
Both numbers are already fractions.
- 2Find a common denominator
LCM(9, 9) = 9
The least common multiple of 9 and 9 is 9, so use 9 as the denominator.
- 3Rewrite both fractions
2×1/9×1 + 4×1/9×1 = 2/9 + 4/9
Multiply top and bottom of each fraction so the denominators match.
- 4Add the numerators
2 + 4 = 6 → 6/9
Keep the denominator unchanged.
- 5Simplify
6/9 ÷ 3/3 = 2/3
The greatest common factor of 6 and 9 is 3; divide both by it.
- 6Decimal and percent
2/3 ≈ 0.666667 = 66.6667%
Different denominators: use the least common denominator
Find the least common multiple of the two denominators. Rewrite each fraction with that number as its denominator by multiplying top and bottom by the same whole number. Then add as before.
Unlike denominators: 2 and 3
- Decimal: 0.833333
Step-by-step working 6 steps
- 1Write the problem
1/2 + 1/3
Both numbers are already fractions.
- 2Find a common denominator
LCM(2, 3) = 6
The least common multiple of 2 and 3 is 6, so use 6 as the denominator.
- 3Rewrite both fractions
1×3/2×3 + 1×2/3×2 = 3/6 + 2/6
Multiply top and bottom of each fraction so the denominators match.
- 4Add the numerators
3 + 2 = 5 → 5/6
Keep the denominator unchanged.
- 5Simplify
5 and 6 share no common factor other than 1, so the fraction is already in lowest terms.
- 6Decimal and percent
5/6 ≈ 0.833333 = 83.3333%
Subtracting works the same way
Match the denominators, then subtract the numerators. The result can be negative — 3/4 − 5/6 comes out as −1/12 because 5/6 is the larger fraction.
Subtraction with a negative result
- Decimal: −0.083333
Step-by-step working 6 steps
- 1Write the problem
3/4 − 5/6
Both numbers are already fractions.
- 2Find a common denominator
LCM(4, 6) = 12
The least common multiple of 4 and 6 is 12, so use 12 as the denominator.
- 3Rewrite both fractions
3×3/4×3 − 5×2/6×2 = 9/12 − 10/12
Multiply top and bottom of each fraction so the denominators match.
- 4Subtract the numerators
9 − 10 = −1 → −1/12
Keep the denominator unchanged.
- 5Simplify
1 and 12 share no common factor other than 1, so the fraction is already in lowest terms.
- 6Decimal and percent
−1/12 ≈ −0.083333 = −8.3333%
Mixed numbers
Convert each mixed number to an improper fraction, add, then convert back. 1 1/2 + 2 3/4 equals 17/4, which is 4 1/4 as a mixed number.
Mixed numbers
- Mixed number: 4 1/4
- Decimal: 4.25
Step-by-step working 7 steps
- 1Write the problem
3/2 + 11/4
Convert mixed numbers and decimals to fractions first: 1 1/2 = 3/2; 2 3/4 = 11/4.
- 2Find a common denominator
LCM(2, 4) = 4
The least common multiple of 2 and 4 is 4, so use 4 as the denominator.
- 3Rewrite both fractions
3×2/2×2 + 11×1/4×1 = 6/4 + 11/4
Multiply top and bottom of each fraction so the denominators match.
- 4Add the numerators
6 + 11 = 17 → 17/4
Keep the denominator unchanged.
- 5Simplify
17 and 4 share no common factor other than 1, so the fraction is already in lowest terms.
- 6Write as a mixed number
17/4 = 4 1/4
4 wholes with 1 left over, out of 4.
- 7Decimal and percent
17/4 ≈ 4.25 = 425%
Common mistakes
Adding the tops and the bottoms
1/2 + 1/3 ≠ 2/5. Only numerators are added, and only after the denominators match.
Using a common multiple but forgetting to scale the numerator
If you change the denominator by multiplying by 3, the numerator must be multiplied by 3 as well.
Leaving the answer unsimplified
Check whether the numerator and denominator share a factor. The simplify calculator will do it for you.
Frequently asked questions
Do I always need the least common denominator?
No — any common multiple works. The least one just gives the smallest numbers to deal with.
How do I add three fractions?
Find the LCM of all three denominators, convert each fraction, and add the three numerators together.
