TheCalculatory
Math · Everyday Arithmetic Formula verified

Fraction Calculator

Enter two fractions and pick an operation. The result is simplified to lowest terms automatically, with every step shown underneath.

Operation
On-screen number pad
Tap to type into the highlighted field
Result, simplified
5/6
1/2 + 1/3
Decimal equivalent
0.8333
Unsimplified
5/6

Step-by-step proofCheck by hand

How each operation works

Adding or subtracting fractions first converts both to a common denominator, by cross-multiplying, then adds or subtracts the numerators. Multiplying just multiplies numerators together and denominators together. Dividing flips the second fraction and multiplies — dividing by a number and multiplying by its reciprocal are the same operation.

Every result is reduced to lowest terms using the greatest common divisor, computed with exact integer arithmetic (the Euclidean algorithm) rather than decimals, so nothing rounds before the final simplification.

The formulas

a/b + c/d = (ad + bc) / bd  ·  a/b × c/d = ac / bd  ·  a/b ÷ c/d = ad / bc

Worked example

1/2 + 1/3: cross-multiplying gives (1×3 + 1×2) / (2×3) = 5/6, which is already in lowest terms since 5 and 6 share no common factor — equal to 0.8333 as a decimal.

Frequently asked questions

Why do I need a common denominator to add or subtract fractions?
A fraction's denominator says how many equal pieces the whole is cut into, so pieces from different-sized cuts can't be counted together directly — a third and a quarter aren't the same size of piece. Converting both fractions to the same denominator first makes the pieces the same size, so the numerators can simply be added or subtracted.
Why doesn't multiplying fractions need a common denominator?
Multiplication asks a different question — a fraction of a fraction — rather than combining pieces of the same size, so the numerators and denominators just multiply straight across: two-thirds of three-quarters is (2×3)/(3×4). No common denominator is needed, because nothing is being added.
Why does dividing by a fraction mean flipping and multiplying?
Dividing by a number asks how many times it fits into another. Dividing by a fraction like 1/2 asks how many halves fit — which is the same as multiplying by 2, its reciprocal. Flipping the second fraction and multiplying always gives the identical answer, through simpler arithmetic.
What does 'simplified to lowest terms' mean?
It means the numerator and denominator share no common factor greater than 1 — 6/8 and 3/4 represent the same amount, but 3/4 is the simplified form, because 3 and 4 share no common factor. Dividing both numbers by their greatest common divisor always produces the simplified version.
Sources and method
  • Standard fraction arithmetic (addition, subtraction, multiplication, division via the reciprocal) — general published mathematics.
  • Results are computed with exact integer arithmetic and simplified via the Euclidean algorithm, not with floating-point decimals, so no rounding error enters before the final display.
Last reviewed: 19 Sep 2026 Sources last verified: 19 Sep 2026 Results use the assumptions explained on this page. Report an error How we check calculations

Related calculators

Percentage Calculator Percent of, part-to-whole, percentage change and adjustments. 2 inputsOpen → Average Calculator Mean, median, mode and range from any list of numbers. 1 inputOpen → Square Root Calculator Square roots and nth roots, with simplified radical form for integers. 2 inputsOpen → Discount Calculator Sale price, amount saved, or work backward from a sale price to the original. 2 inputsOpen →