Fraction calculator: add, subtract, multiply and divide

How to work with fractions

Addition and subtraction require the two fractions to speak the same language, that is to share a denominator. You find the lowest common denominator — the least common multiple of the two denominators — rewrite both fractions on that base and add the numerators only. For 1/2 + 1/3 the lowest common denominator is 6: the fractions become 3/6 and 2/6, and the sum is 5/6.

Multiplication and division are simpler and need no common denominator. In a product you multiply numerators together and denominators together. In a division you multiply the first fraction by the reciprocal of the second, that is you flip the second and multiply: 3/4 ÷ 2/5 becomes 3/4 × 5/2 = 15/8.

Reducing to lowest terms means dividing numerator and denominator by their greatest common divisor. 8/12 has a GCD of 4 and reduces to 2/3: it is the same quantity written as compactly as possible, and it is the form in which an answer is normally expected. When the numerator exceeds the denominator the fraction is called improper and can be rewritten as a mixed number: 7/3 becomes 2 and 1/3.

Converting a decimal to a fraction is exact for terminating decimals — 0.375 is 375/1000, that is 3/8 — and uses the nines rule for recurring ones: 0.8333… is recognised as 5/6. This is why a fraction is sometimes preferable to a decimal: 1/3 is exact, whereas 0.333 is an approximation that carries error into every later calculation.

Common mistakes

Adding numerators and denominators separately: 1/2 + 1/3 is not 2/5. You need the common denominator first.
Cancelling across a sum: in (a + b)/b you cannot cancel the b, because cancelling only works between factors of a product.
Forgetting to flip the second fraction in a division, or flipping the first instead of the second.
Leaving the result unreduced: 6/8 is correct but incomplete, the expected form is 3/4.
Treating a recurring decimal as if it terminated: rounding 1/3 to 0.33 introduces an error that grows through later multiplications.

Frequently asked questions

How do you add two fractions with different denominators?

Find the lowest common denominator, rewrite both fractions on that base by multiplying numerator and denominator by the same factor, then add the numerators and keep the denominator. Finally reduce the result to lowest terms.

How do you divide one fraction by another?

Multiply the first by the reciprocal of the second: a/b ÷ c/d = a/b × d/c = (a × d) / (b × c). The operation is undefined if c is zero, since that would amount to dividing by zero.

What is a mixed number?

It is an improper fraction written as a whole part plus a proper fraction: 7/3 equals 2 plus 1/3, written 2 1/3. The two forms represent the same number.

How do you turn a decimal into a fraction?

For a terminating decimal, write the number without the point over 10 raised to the number of decimal places, then simplify: 0.375 = 375/1000 = 3/8. For a recurring decimal, use the nines rule: 0.8333… = 5/6.

How do you reduce a fraction to lowest terms?

Divide the numerator and the denominator by their greatest common divisor. For 18/24 the GCD is 6, so the reduced fraction is 3/4. If the GCD is 1 the fraction is already in lowest terms.

How this calculation works

Addition and subtraction: a/b ± c/d = (a × m/b ± c × m/d) / m, where m is the least common multiple of b and d. Product: a/b × c/d = (a × c) / (b × d). Quotient: a/b ÷ c/d = (a × d) / (b × c). The result is reduced by dividing numerator and denominator by their greatest common divisor, found with Euclid's algorithm, and the sign is always carried by the numerator.

Percentages

Percentage of a total, increases, discounts, percentage change and reverse percentage.

Proportions

Missing term, fourth proportional, geometric mean and third proportional.

GCD and LCM

Greatest common divisor and least common multiple with prime factorization.

Prime factorisation

Prime factors of a number up to 30 digits, with the division ladder, divisors and a primality test.