Subtracting Fractions Calculator

Subtracting fractions works exactly like adding them, with two differences that matter: the order of the two numbers changes the answer, and the answer is allowed to be negative. This calculator prints every intermediate line and keeps the sign visible instead of quietly flipping the question around.

The formula and where it comes from

A denominator is not a number being subtracted — it is a label saying how the whole was cut up. Three quarters minus one quarter is two quarters because every piece named in the difference is the same size. Halves and thirds are not the same size, so 1/2 - 1/3 is not 0/1: before anything can be taken away, 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 = 1/6.

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. Subtract the second numerator from the first. The denominator does not change; if the second numerator is larger, the result is negative and keeps its minus sign.
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 subtractions

Where fraction subtraction is used

• Cooking: a jar holds 3/4 of a cup and a recipe takes 1/2 — what is left is 1/4 of a cup, not half of what you started with.
• Woodworking: cutting 1/4 inch off a 7/8 inch board leaves 5/8 inch, which is why the common denominator of 8 shows up on every tape measure.
• Time: a task planned for 2 1/3 hours that finished half an hour early took 1 5/6 hours.
• Money and shares: an owner holding 2/3 who sells a 1/3 share is left with 1/3 — the same denominator on both sides makes the subtraction immediate.
• Temperature and levels: a tank at 1/4 full that should be at 1/2 is 1/4 short, and the page states that as a negative difference when the numbers are entered the other way round.

Limits and things to watch

• Order changes the answer. 3/4 - 1/2 = 1/4, but 1/2 - 3/4 = -1/4. Subtraction is not commutative, so these are two different pages with two different answers — unlike addition, where the pair has a single page.
• A negative answer is a real answer. 1/4 - 1/2 = -1/4. Swapping the numbers to avoid the minus sign answers a different question; the size is the same, the sign is not.
• Denominators are never subtracted. 1/2 - 1/3 is 1/6, not 1/1 and not 0. The denominator names the size of a part, and that size does not change when parts are taken away.
• Borrowing is only needed with mixed numbers. 3 1/4 - 1/2 turns into 13/4 - 2/4 = 11/4 with no borrowing at all once both numbers are improper fractions.
• The decimal form is often not exact. 1/2 - 1/3 = 1/6 is 0.1666… 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 1/6; the page reduces first and says so, so the two pages agree instead of contradicting each other.