Fraction Calculator
Add, subtract, multiply and divide fractions and mixed numbers, or simplify one — with the answer as a reduced fraction, a mixed number, a decimal (including repeating decimals) and a percent, and every step shown.
Written by Suhaib Hassan. The rules are standard arithmetic and the worked examples are re-derived by hand. Our methodology.
It uses arbitrary-size integers, finds the lowest common denominator, combines the numerators, then reduces with Euclid’s greatest-common-divisor algorithm. No rounding occurs until the decimal display.
Fractions are exact, decimals are not. This page keeps the exact answer and shows every step, including how a repeating decimal repeats.
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Number line
Where the fractions and the result sit between whole numbers.
Step-by-step working
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Rules for fraction arithmetic
The four operations and simplifying.
| Operation | Rule | Example | Condition |
|---|---|---|---|
| Add | a/b + c/d = (ad + bc) / bd | 3/4 + 2/3 = 17/12 | b, d ≠ 0; the lowest common denominator keeps numbers small |
| Subtract | a/b − c/d = (ad − bc) / bd | 3/4 − 2/3 = 1/12 | The result can be negative |
| Multiply | a/b × c/d = ac / bd | 3/4 × 2/3 = 1/2 | Cancel common factors first to keep numbers small |
| Divide | a/b ÷ c/d = ad / bc | 3/4 ÷ 2/3 = 9/8 | c ≠ 0: flip the divisor and multiply |
| Simplify | a/b = (a÷g) / (b÷g) | 18/24 = 3/4 | g = GCD(a, b) |
Worked examples
Adding lengths
A 1 5/8″ board plus a 3/32″ gasket.
52/32 + 3/32 = 55/32
= 1 23/32 inches
Combining volumes
2/5 mL of one solution plus 3/8 mL of another.
16/40 + 15/40 = 31/40 mL
Scaling a recipe
A recipe needs 3/4 cup of sugar; you make 2 2/3 batches.
= 2 cups
What is a fraction?
A fraction a/b represents a parts out of b equal parts, or the quotient a ÷ b. It is proper if a < b, improper if a ≥ b, and a mixed number combines a whole number with a proper fraction (1 5/12 = 17/12). Two fractions are equivalent if they have the same value: 2/4 = 1/2.
How to simplify
Divide the numerator and denominator by their greatest common divisor (GCD), found with Euclid’s algorithm. For 17/12: GCD(17, 12) = GCD(12, 5) = GCD(5, 2) = GCD(2, 1) = 1, so 17/12 is already in lowest terms.
Repeating decimals
A fraction in lowest terms has a terminating decimal exactly when its denominator has no prime factors other than 2 and 5. Otherwise the decimal repeats; the repeating block is at most (denominator − 1) digits long. For example 17/12 = 1.41666… = 1.41(6).
By Suhaib Hassan · Reviewed 28 September 2026 · How CalculatePilot verifies formulas →
Sources
- Euclid, Elements, Book VII, Propositions 1–2 — the Euclidean algorithm for the greatest common divisor.
- Knuth, D. E. (1997). The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed., section 4.5 (rational arithmetic).
- ISO 80000-2:2019, Quantities and units — Part 2: Mathematics.
Frequently asked questions
Why use the lowest common denominator instead of just multiplying the denominators?
Any common denominator works, but the lowest one keeps the numbers small and often skips a later simplification step. For 5/6 + 1/4, multiplying gives 26/24 which must be reduced to 13/12; the LCD 12 gives 13/12 directly.
How can I tell whether a decimal will terminate or repeat?
Reduce the fraction, then factor the denominator. If it contains only 2s and 5s, the decimal terminates (3/8 = 0.375). Any other prime factor makes it repeat (1/3 = 0.(3), 1/7 = 0.(142857)).
How do negative fractions work?
Only the fraction as a whole carries a sign: −3/4 = 3/(−4) = −(3/4). This calculator keeps the denominator positive and puts the sign on the numerator (or in front of a mixed number).
How do I convert a mixed number to an improper fraction?
Multiply the whole number by the denominator, add the numerator, and keep the denominator: 2 1/3 = (2 × 3 + 1)/3 = 7/3. Going the other way, divide and keep the remainder: 7/3 = 2 remainder 1, so 2 1/3.
Why can’t the denominator be zero?
Division by zero is undefined. A denominator of 0 would mean splitting something into zero parts. Dividing by a fraction whose numerator is 0 is likewise undefined.
What is a continued fraction?
It writes a number as a whole part plus 1 over another whole part plus 1 over …: 17/12 = 1 + 1/(2 + 1/(2 + 1/2)), written [1; 2, 2, 2]. The quotients are exactly the ones produced by Euclid’s algorithm.
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