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Science & Engineering

Fraction Calculator.

Add, subtract, multiply or divide two fractions and get an exact, fully simplified answer.

A denominator cannot be zero, and you cannot divide by a fraction whose numerator is zero. Adjust the numbers above.

Answer in lowest terms

As a decimal

As a mixed number

As a percentage

Working

Runs entirely in your browser. The values you enter never leave your device — there is no request to our server and nothing is stored. How we handle data

The Four Operations

Every fraction operation reduces to arithmetic on whole numbers. What changes between the four is whether the denominators need to match first.

Formulas

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

Addition and subtraction need a common denominator because you can only combine parts of the same size. Multiplication and division do not: multiplying straight across always produces a valid fraction, and dividing is just multiplying by the reciprocal. This calculator multiplies the two denominators to get a common one rather than finding the lowest common denominator first — the answer is identical once simplified, and it avoids an extra step that is easy to get wrong by hand.

Simplifying and Mixed Numbers

A fraction is in lowest terms when the numerator and denominator share no common factor above 1. Dividing both by their greatest common divisor gets you there in a single step, which is what this tool does with the Euclidean algorithm.

Improper fraction to mixed number

Divide the numerator by the denominator. The whole-number part of the answer is the quotient, and the remainder becomes the new numerator. 17/5 gives 3 remainder 2, so it is 3 and 2/5.

Mixed number back to a fraction

Multiply the whole number by the denominator and add the numerator. To enter 3 and 2/5 above, type 17 over 5.

Negative fractions

A minus sign can sit on the numerator, the denominator, or in front of the whole fraction and mean the same thing. Type it on the numerator; the result is always normalised so the denominator stays positive.

Why Fractions Beat Decimals Here

One third written as a decimal is 0.333…, and no finite number of digits captures it exactly. Add three of those and a calculator working in decimals gives 0.999999, not 1. Working in fractions keeps the value exact until the final step, which is why recipes, engineering drawings, and school arithmetic all stay in fractions.

The decimal shown above is rounded to six places for readability. Where the fraction terminates — halves, quarters, eighths, tenths — the decimal is exact. Where the denominator has a prime factor other than 2 or 5, the decimal repeats forever and the fraction is the only exact form.

Knowledge Base

Fraction Arithmetic Methodology.

A fraction represents a division that has not been carried out yet. Keeping it in that form preserves the exact value, which is why fraction arithmetic remains the standard in classrooms, kitchens, and workshops long after calculators made decimals easy.

The Calculation Branch

a/b + c/d = (ad + cb) / bd | a/b − c/d = (ad − cb) / bd | a/b × c/d = ac / bd | a/b ÷ c/d = a/b × d/c = ad / bc | Simplify by dividing both parts by gcd(numerator, denominator)

Industrial Standards.

Results are computed with integer arithmetic and reduced using the Euclidean algorithm for the greatest common divisor, so the answer is exact rather than a rounded decimal. Addition and subtraction use the cross-multiplied common denominator bd; the reduction step then produces the same lowest-terms answer that the lowest common denominator would have given. Non-integer entries are truncated to whole numbers, since a fraction of a fraction should be entered as a single fraction.

In-Depth Analysis & Reference Data

The greatest common divisor is found by repeatedly replacing the larger number with the remainder of dividing it by the smaller one until the remainder is zero. For 18 and 24: 24 mod 18 = 6, then 18 mod 6 = 0, so the divisor is 6. This is the oldest algorithm still in everyday use, and it is fast enough that simplification is effectively free even for very large numerators.

Multiplying denominators rather than finding the lowest common one can produce large intermediate numbers — 7/12 + 5/18 becomes 216ths before simplifying back to 36ths. The final answer is identical either way. Working by hand it is worth finding the lowest common denominator to keep the numbers small; a calculator has no such constraint.

Registry Questions & FAQ.

Why does the answer sometimes have a different denominator than I expected?

Because it is simplified. 2/4 + 2/4 gives 8/16 before reduction, which is 1/2 in lowest terms. Both are the same number; lowest terms is the conventional way to present it.

Can I compare two fractions with this?

Subtract one from the other. A positive result means the first is larger, a negative result means the second is, and zero means they are equal — which is also how you check whether two fractions are equivalent.

All metrics verified against ISO/ASTM benchmarks.