TanodTools

Fraction calculator

Calculate with fractions, mixed numbers and decimals and get the exact answer, in lowest terms, with the steps.

Runs entirely in your browser

What do you want to do?
Operation

Formats: 3/4, 2 1/2 (two and a half), -1 1/2 (minus one and a half), 0.75, 1.2e-3, 0.(3) for a repeating decimal.

Arithmetic is exact, using whole numbers of any length, so nothing is rounded. A decimal expansion is shown to 300 digits at most; a repeating block longer than that is cut off and marked.

How to calculate with fractions

  1. Type the first fraction, such as 1/2 or 2 1/2, choose an operation, and type the second. Decimals such as 0.75 work too.
  2. Read the answer in lowest terms, as a mixed number and as a decimal, with any repeating digits in brackets.
  3. Open the steps to see the common denominator, or the cross-multiplication for division. Switch to Simplify or convert to reduce one fraction or turn a decimal into a fraction.

Why fractions are worth keeping exact

A decimal such as 0.333333 only approximates one third, and every operation on approximations adds a little more error. A fraction is exact: 1/3 + 1/6 is precisely 1/2. This calculator therefore keeps every value as a pair of whole numbers, numerator and denominator, and reduces the pair by dividing both by their greatest common divisor after each step. Only at the end does it work out the decimal, by long division, and say which digits repeat.

Adding and subtracting needs a shared denominator. The least common denominator is the smallest number that both denominators divide into: for 1/4 and 1/6 it is 12, so the sum is 3/12 + 2/12 = 5/12. Multiplying is more direct, numerator times numerator over denominator times denominator. Dividing turns into multiplying by the reciprocal of the second fraction, which is why you cannot divide by a fraction equal to zero.

A mixed number such as 3 1/7 is just a whole number plus a proper fraction, and it equals the improper fraction 22/7. Converting back and forth is a matter of multiplying the whole part by the denominator and adding the numerator. Negative values keep their sign on the whole: −3 1/7 means −(3 + 1/7), the same as −22/7.

Tips

Questions

How do I enter a mixed number?

Type the whole number, a space, then the fraction: 2 1/2 means two and a half. A minus sign in front applies to the whole mixed number, so -1 1/2 is −3/2, that is −(1 + 1/2), and not −1 + 1/2. The result is always shown with the sign on the whole value, for example −1 1/2.

What decimals can I type?

Ordinary decimals such as 0.75 or .5, exponent forms such as 1.2e-3 (which is 3/2500), and repeating decimals with the repeating digits in brackets: 0.(3) is 1/3 and 0.1(6) is 1/6. Each is converted to an exact fraction, so 0.1 is exactly 1/10 here and 0.1 + 0.2 is exactly 3/10.

How are the steps worked out?

To add or subtract, the calculator finds the least common denominator, rewrites both fractions over it, combines the numerators and then reduces. For multiplication it multiplies across, and for division it multiplies by the reciprocal, which is the same as cross-multiplying: a/b ÷ c/d = (a × d) ÷ (b × c).

Why do some fractions have repeating decimals?

A fraction in lowest terms ends after a few digits only when its denominator has no prime factors other than 2 and 5. Otherwise the long division returns to a remainder it has seen before, and the digits repeat from there: 1/7 is 0.(142857), with a six-digit block, and 1/6 is 0.1(6). Brackets show the repeating block.

What happens with zero in a denominator?

A fraction such as 3/0 is rejected with a message, since no number multiplied by 0 gives 3. Dividing by a fraction whose value is 0, for example 1/2 ÷ 0, is rejected as well. Zero in the numerator is fine: 0/5 is 0.

Can it handle very large numbers?

Yes. The calculator uses arbitrary-length integers, so 123456789012345678901/2 + 1/3 comes out as exactly 370370367037037036705/6, with no rounding to 15 digits as there would be in a calculator that uses floating point. For decimal-style work with trig and logs, use the scientific calculator.