Adding and subtracting needs a common denominator
Fractions can only be added when the pieces are the same size, so rewrite both over a shared denominator first.
3/4 + 2/5: the denominators 4 and 5 give 20.
- 3/4 = 15/20
- 2/5 = 8/20
- 15/20 + 8/20 = 23/20, or 1 3/20.
Subtraction is identical: 3/4 − 2/5 = 15/20 − 8/20 = 7/20.
Multiplying the two denominators always works. The smallest common denominator is their least common multiple, which keeps the numbers down: for 1/6 + 1/4 that is 12, not 24.
Multiplying is the easy one
Multiply across the top and across the bottom. No common denominator needed.
2/3 × 3/5 = 6/15 = 2/5.
Dividing means multiplying by the reciprocal
Flip the second fraction and multiply.
2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6.
This is why dividing by a fraction smaller than 1 makes the answer bigger: 2 ÷ 1/2 = 2 × 2 = 4.
Simplifying
Divide the top and bottom by their greatest common divisor. For 18/24 that is 6, giving 3/4. If you cannot see the GCD, divide by any shared factor you do see and repeat: 18/24 → 9/12 → 3/4.
Mixed numbers
Turn them into improper fractions before doing anything else. 2 1/3 = (2 × 3 + 1)/3 = 7/3. To go back, divide: 23/20 is 20 into 23 once with 3 left, so 1 3/20.
Common mistakes
- Adding the denominators. 1/2 + 1/3 is not 2/5. A half plus a third is more than a half, and 2/5 is less.
- Finding a common denominator to multiply. It is not wrong, just unnecessary work.
- Cancelling across a plus sign. In (x + 2)/2 the 2s do not cancel. Cancelling only works across multiplication.
Tools
The Fraction Calculator adds, subtracts, multiplies and divides fractions and mixed numbers, simplifies the result and shows it as a decimal. The Percentage Calculator handles the same comparisons in percent form.