Fraction Calculator

The four rules

Adding and subtracting

Fractions can only be added when they are cut into the same size pieces, so both must first be rewritten over a common denominator. Multiplying each fraction by the other's denominator always achieves that:

Multiplying

Multiplication needs no common denominator at all. Multiply straight across the top and straight across the bottom:

Dividing

Dividing by a fraction is the same as multiplying by its reciprocal - flip the second fraction and multiply. This works because dividing by c/d asks "how many lots of c/d fit in?", which is the same as scaling by d/c.

Worked examples

1. Add 1/2 and 1/3

   1/2 + 1/3

   1. Find a common denominator: 2 × 3 = 6.
   2. Rewrite both: 1/2 = 3/6 and 1/3 = 2/6.
   3. Add the numerators only: 3 + 2 = 5, keeping the denominator 6.
   4. Check whether it reduces - 5 and 6 share no factor, so it is already in lowest terms.

   = 5/6

2. Subtract 5/6 from 3/4

   3/4 − 5/6

   1. Common denominator: 4 × 6 = 24.
   2. Rewrite: 3/4 = 18/24, 5/6 = 20/24.
   3. Subtract: 18 − 20 = −2, giving −2/24.
   4. Reduce by their greatest common divisor, 2: −1/12.

   = −1/12

3. Multiply 2/3 by 3/8

   2/3 × 3/8

   1. Multiply the numerators: 2 × 3 = 6.
   2. Multiply the denominators: 3 × 8 = 24.
   3. That gives 6/24; the greatest common divisor of 6 and 24 is 6.
   4. Reduce: 6 ÷ 6 = 1 and 24 ÷ 6 = 4.

   = 1/4

4. Divide 3/4 by 2/5

   3/4 ÷ 2/5

   1. Flip the second fraction: 2/5 becomes 5/2.
   2. Change the division to multiplication: 3/4 × 5/2.
   3. Multiply across: (3×5) / (4×2) = 15/8.
   4. 15 and 8 share no common factor, so this is final. As a mixed number it is 1⅞.

   = 15/8

Reducing to lowest terms

A fraction is in lowest terms when its numerator and denominator share no common factor above 1. To reduce, divide both by their greatest common divisor (GCD). The GCD is found with the Euclidean algorithm: repeatedly replace the larger number with the remainder of dividing it by the smaller, until the remainder is zero.

1. To reduce 84/126, first find gcd(84, 126).
2. 126 mod 84 = 42.
3. 84 mod 42 = 0, so the GCD is 42.
4. Divide both parts: 84 ÷ 42 = 2, 126 ÷ 42 = 3.
5. The reduced fraction is 2/3.

Fraction, decimal and percentage equivalents

That last point is worth dwelling on. 1/8 terminates because 8 = 2³. 1/3 does not, because 3 is neither 2 nor 5. This is a property of base 10, not of the fraction - in base 3, one third would be a clean 0.1. The same logic in base 2 is exactly why 0.1 + 0.2 misbehaves in binary floating point.

Frequently Asked Questions