Fraction Calculator

The four operations

Multiplication is the easy one: multiply the numerators and multiply the denominators. Two thirds times three quarters is six twelfths, which simplifies to one half.

Division inverts the second fraction and multiplies. Two thirds divided by three quarters is two thirds times four thirds, which is eight ninths.

Addition and subtraction need a common denominator first. One third plus one quarter requires twelfths: four twelfths plus three twelfths is seven twelfths. The lowest common denominator keeps the numbers small, and any common denominator works.

Simplifying

A fraction is simplified by dividing numerator and denominator by their greatest common divisor. Twelve eighteenths divides by six to give two thirds.

Euclid's algorithm finds the greatest common divisor efficiently: repeatedly replace the larger number by its remainder when divided by the smaller, until one is zero. It is over two thousand years old and remains the standard method.

Simplifying is conventional rather than required. Twelve eighteenths and two thirds are the same number, and the simplified form is preferred because it is easier to compare and to work with.

Mixed numbers and improper fractions

An improper fraction has a numerator larger than its denominator, such as seven thirds. A mixed number expresses the same value as a whole number and a proper fraction: two and one third.

Converting to a mixed number divides and keeps the remainder. Converting back multiplies the whole number by the denominator and adds the numerator.

Calculation is easier in improper form, since mixed numbers have to be converted before multiplying or dividing anyway. Presentation is usually clearer as a mixed number, which is why the convention is to calculate in one form and present in the other.

Fractions and decimals

Every fraction is a decimal, terminating or repeating. A fraction terminates when its denominator, in lowest terms, has no prime factors other than 2 and 5.

One eighth is 0.125 and terminates because 8 is 2 cubed. One third is 0.333 recurring because 3 is neither 2 nor 5. One seventh repeats with a six-digit cycle.

Which is why some decimal arithmetic in computers is inexact: 0.1 cannot be represented exactly in binary floating point for the same reason one third cannot be represented exactly in decimal. Working in fractions avoids the problem entirely, which is why exact arithmetic libraries use them.

Where fractions are the natural form

Cooking uses them because measures are physically divided: half a cup, three quarters of a teaspoon. Scaling a recipe by two thirds is fraction arithmetic whether or not anyone calls it that.

Imperial construction measurements are fractional: sixteenths and thirty-seconds of an inch. Tape measures are marked that way and converting to decimals loses the correspondence with the tool.

Music uses them for note durations and time signatures. Probability uses them because a probability is a ratio. In each case the fraction is closer to what is being described than a decimal approximation would be.

Ratios and proportions are fractions in different clothing, which is worth noticing because it makes a lot of practical arithmetic easier. A 3:2 ratio is three fifths and two fifths of the whole, not three halves, and mixing those two readings is behind a good deal of confusion about mixing concrete, diluting solutions and scaling recipes. Once a ratio is converted to fractions of the total, scaling it to any quantity is a single multiplication.