Fraction calculator

How it works

Adding and subtracting need the same denominator. The common denominator is found by cancelling one denominator against the other first, rather than simply multiplying them. Adding 1/6 and 1/10 the direct way gives 16/60, while this way gives 8/30. Both then reduce to 4/15, but the second working is half the size.

Multiplying is the simplest: numerator times numerator, denominator times denominator. Dividing is the same with the second fraction turned upside down, since dividing by a fraction is the same as multiplying by its reciprocal.

Before multiplying, the numerator of one fraction is cancelled against the denominator of the other. That keeps the numbers small, which both gives tidier working and pushes back the point where they grow too large to be exact.

The answer is always reduced by the greatest common divisor, and a minus sign in the denominator is moved up to the numerator so the form is unambiguous.

A fraction has a terminating decimal exactly when its reduced denominator contains only the factors two and five. 3/8 stops after three decimal places, while 1/3 runs on forever. The line under the answer says which it is.

If any step grows too large to be exact, no answer is given at all. A fraction with a wrong digit in the denominator looks like an answer but is not one.

Worked example

5/6 + 7/10: the greatest common divisor of 6 and 10 is 2, so the common denominator is 6 × 10 / 2 = 30. The fractions become 25/30 and 21/30, the sum is 46/30, and the tool reports “Reduced by 2.” The answer is 23/15, as a mixed number 1 8/15, and the decimal 1.533333 is marked as rounded, since 15 contains the factor 3.

4/9 × 15/8: before multiplying, 4 is cancelled against 8 by 4 and 15 against 9 by 3, so the calculation becomes (1 × 5) / (3 × 2) = 5/6. The working shows the uncancelled product 60/72 and that it was reduced by 12.

The greatest common divisor

The divisor is found with Euclid’s algorithm. In the Elements, Book VII, Proposition 2, the smaller number is subtracted from the larger again and again until what is left measures the number before it. The tool does the same with division: divide the larger number by the smaller, keep the remainder and repeat until the remainder is zero. For 46 and 30 that gives 46 = 1 × 30 + 16, 30 = 1 × 16 + 14, 16 = 1 × 14 + 2 and 14 = 7 × 2. The last remainder that is not zero, 2, is the divisor.

Input and limits

The fields take whole numbers, fractions such as 3/4 and mixed numbers such as 1 3/8, with the minus sign before the whole number: −1 3/8 is −11/8. A minus sign in the denominator is accepted too. Decimals such as 0.5 are not; write 1/2.

Every numerator and denominator must fit in the browser’s exact integers, that is, be at most 9,007,199,254,740,991, or 2⁵³ − 1. 1/99999989 + 1/99999971 gives no answer, because the common denominator comes to almost 10¹⁶. The decimal under the answer is shown to at most six places and the percentage to at most four. The decimal is marked exact only when the whole expansion fits in those six places, so 1/128 = 0.0078125 is shown as 0.007813 and marked rounded.

Sources

• Euclid – Elements, Book VII, Proposition 2 (ed. David E. Joyce, Clark University)
• Ecma International – ECMA-262, ECMAScript Language Specification

How the tools are checked