Fraction Calculator

A fraction is a top number (the numerator) over a bottom number (the denominator): 3/4 means 3 parts out of 4 equal parts. To add or subtract fractions, the parts must be the same size, so you first give them a common bottom. To multiply, you multiply straight across. To divide, you flip the second fraction and multiply. At the end you divide the top and bottom by their greatest common divisor (GCD) to get the simplest form.

The formula

mixed number w n/d = (w × d + n) / d
a/b + c/d = (a × L ÷ b + c × L ÷ d) / L, where L = lowest common denominator of b and d
a/b − c/d = (a × L ÷ b − c × L ÷ d) / L
a/b × c/d = (a × c) / (b × d)
a/b ÷ c/d = a/b × d/c
simplest form = (top ÷ GCD) / (bottom ÷ GCD)

Worked example

1 1/2 + 5/6. First turn the mixed number into an improper fraction: 1 1/2 = (1 × 2 + 1)/2 = 3/2. The lowest common denominator of 2 and 6 is 6, so 3/2 = (3 × 3)/(2 × 3) = 9/6. Now add the tops: 9/6 + 5/6 = 14/6. The GCD of 14 and 6 is 2, so 14/6 = (14 ÷ 2)/(6 ÷ 2) = 7/3. As a mixed number, 7 ÷ 3 = 2 remainder 1, so it is 2 1/3. As a decimal it is 2.333… (the 3 repeats for ever), which is 233.33%.

Finding the GCD (Euclid’s method)

Divide the bigger number by the smaller and keep the remainder. Then divide the smaller number by that remainder, and repeat until the remainder is 0. The last number you divided by is the GCD. For 12/16: 16 = 1 × 12 + 4, then 12 = 3 × 4 + 0, so the GCD is 4 and 12/16 = 3/4. The lowest common denominator of two bottoms is their product divided by their GCD: for 4 and 6 it is 24 ÷ 2 = 12.

Decimals and repeating decimals to fractions

A decimal that stops goes over 1 followed by as many zeros as it has decimal places: 0.375 = 375/1000 = 3/8 (the GCD is 125). For a repeating decimal, call it x and shift it so the repeats line up. For 0.1666…: 100x = 16.666… and 10x = 1.666…, so 90x = 15 and x = 15/90 = 1/6. In Advanced, type 0.1(6) or put 6 in the repeating box. A percent is a number over 100: 37.5% = 37.5/100 = 3/8.

Rounding to tape-measure fractions

To round a decimal to the nearest 1/16 (common on imperial rulers and drill bits), multiply by 16, round to a whole number and put it over 16: 0.37 × 16 = 5.92, which rounds to 6, and 6/16 = 3/8. The “Closest with bottom up to …” option finds the nearest fraction with a small bottom: 3.14159265 is closest to 22/7 with a bottom up to 10 and 311/99 with a bottom up to 100.

Comparing fractions

Give them all the same bottom and compare the tops. For 1/2, 3/5 and 2/3 the lowest common denominator is 30: 15/30, 18/30 and 20/30, so 1/2 < 3/5 < 2/3. For just two fractions you can cross-multiply: 1/2 vs 2/3 compares 1 × 3 = 3 with 2 × 2 = 4, so 1/2 is smaller.

Typing a long sum

• A fraction typed with no spaces round the slash, like 1/2, counts as one number, so 6 ÷ 1/2 = 12.
• A slash with spaces round it divides in order: 6 / 1 / 2 = 3.
• Write mixed numbers with a space: 2 1/4. Negative numbers: −3/4.
• Powers: (2/3)^2 = 4/9, and 2^−1 = 1/2. Percent: 50% of 3/4 = 3/8.
• The working shows each step in the order it is done: brackets, then powers, then × and ÷, then + and −.

Common fractions

All answers are exact: the calculator works with whole numbers of any size (up to 3,000 digits), so there are no rounding errors until you choose a number of decimal places.