Adding Fractions Calculator

Adding fractions has one rule and one catch: the rule is that numerators add up, and the catch is that they only add up once both fractions describe parts of the same size. This calculator does the sizing first and prints every intermediate line, so the answer can be checked rather than trusted.

The formula and where it comes from

A denominator is not a number being added — it is a label saying how the whole was cut up. Two quarters plus one quarter is three quarters because every piece named in the sum is the same size, so counting them is meaningful. Halves and thirds are not the same size, which is why 1/2 + 1/3 is not 2/5: before anything can be counted, both fractions are rewritten over a common denominator, and the smallest one that works is the least common multiple of the two denominators. For 2 and 3 that is 6, so 1/2 + 1/3 becomes 3/6 + 2/6 = 5/6.

Step-by-step guide

1. Simplify any fraction that can be simplified. 2/4 becomes 1/2 — the value does not change, and the common denominator that follows stays smaller.
2. Write both numbers as fractions. A whole number goes over 1 (2 = 2/1); a mixed number becomes an improper fraction (2 1/3 = (2 × 3 + 1)/3 = 7/3).
3. Find the least common denominator — the least common multiple of the two denominators.
4. Rewrite each fraction over it, multiplying numerator and denominator by the same number.
5. Add the numerators. The denominator does not change: the size of a part is the same before and after.
6. Reduce the result by the greatest common factor of the two parts, and write it as a mixed number if the numerator is larger than the denominator.

Common fraction additions

Where fraction addition is used

• Cooking: a recipe asking for 1/2 cup of milk and then 1/3 cup more needs 5/6 of a cup in total — not one cup, and not 2/5.
• Woodworking and sewing: a 3/4 inch board glued to a 5/8 inch board makes a panel 1 3/8 inches thick.
• Time: three quarters of an hour plus half an hour is 1 1/4 hours, which is where the common denominator of 4 comes from.
• Money and shares: owning 1/4 of something and inheriting another 1/8 leaves 3/8, the share you can then compare with a half.
• Distance: running 2 1/3 miles in the morning and 1/2 a mile in the evening is 2 5/6 miles for the day.

Limits and things to watch

• Denominators are never added. 1/2 + 1/3 is 5/6, not 2/5. The denominator names the size of a part, and the size does not change when parts are counted.
• Order does not matter. 1/3 + 1/2 and 1/2 + 1/3 are the same question with the same answer, because addition is commutative — unlike subtraction, where 1/2 - 1/3 and 1/3 - 1/2 differ by a sign. That is why this calculator publishes one page per pair.
• The sum often passes 1. 3/4 + 3/4 = 6/4 = 3/2, which is the mixed number 1 1/2. An improper fraction is not a mistake, it is just another way to write the same value.
• The decimal form is often not exact. 1/2 + 1/3 = 5/6 is 0.8333… forever; whenever the reduced denominator has a prime factor other than 2 or 5, the decimal repeats and the page marks it with ≈. The fraction is the exact answer.
• An unreduced input is the same question. 2/4 + 1/3 and 1/2 + 1/3 both equal 5/6; the page reduces first and says so, so the two pages agree instead of contradicting each other.