Fraction Calculator

Six calculators sit above this. The first runs one operation on two fractions and prints every line of the working. The rest handle mixed numbers, reduce a fraction to lowest terms, convert a decimal to a fraction and a fraction back to a decimal, and repeat the same arithmetic on whole numbers far too large for a spreadsheet to hold exactly.

What a fraction actually is

A fraction is a division that has not been carried out yet. The bar means divide. So 3/8 is three divided by eight, and it also names three parts of something that was cut into eight equal parts. The number on top is the numerator, the one underneath is the denominator. The word itself comes from the Latin fractus, broken.

Cut a pie into 8 slices and eat 3 of them. What you ate is 3/8 of the pie and what sits on the plate is 5/8. The two fractions share a denominator because the pie was only cut once. That shared denominator is what makes them easy to compare, and getting to a shared denominator is most of the work in fraction arithmetic.

Proper, improper, mixed

Three names describe the same quantity written three ways:

Form	Example	What it means
Proper	3/8	Numerator smaller than denominator, so the value is under 1
Improper	11/8	Numerator at least as large, so the value is 1 or more
Mixed	1 3/8	Whole part written in front of the leftover fraction

11/8 and 1 3/8 are the same number. Improper form is easier to calculate with, mixed form is easier to picture, and the calculators above give you both. A mixed number carries one sign for the whole thing: -2 3/4 means -(2 + 3/4), which is -11/4, not -2 + 3/4.

Adding and subtracting

Sevenths and eighths cannot be added as they stand, any more than you can add 3 metres to 3 gallons. Both fractions have to be rewritten in the same size of piece first. Multiplying each fraction by the other one's denominator always gets you there:

$$\frac{a}{b}+\frac{c}{d}=\frac{a\times d}{b\times d}+\frac{c\times b}{d\times b}=\frac{ad+cb}{bd}$$

Multiplying a fraction's top and bottom by the same number does not change its value, because you have multiplied by 1 in disguise. 2/7 becomes 16/56 and 3/8 becomes 21/56. Now both are counted in fifty-sixths, the numerators add, and the answer is 37/56.

Subtraction works the same way with one sign changed:

$$\frac{a}{b}-\frac{c}{d}=\frac{ad-cb}{bd}$$

The least common multiple shortcut

Multiplying the denominators together always works, but it can produce numbers larger than you need. Adding 1/4 and 1/6 that way gives twenty-fourths. The smallest denominator that both 4 and 6 divide into is 12, the least common multiple, and working in twelfths keeps the arithmetic smaller and often lands on an answer that is already reduced.

For 7 and 8 the least common multiple is 56, which is just 7 × 8, so the shortcut saves nothing there. It pays off when the denominators share a factor. The step-by-step panel tells you which case you are in.

Multiplying

Multiplication is the easy one. No common denominator is needed. Numerators multiply, denominators multiply:

$$\frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd}$$

So 3/4 × 1/6 = 3/24, which reduces to 1/8. Multiplying by a fraction below 1 makes a number smaller, which surprises people who expect multiplication to increase things. Two thirds of a half is a third.

Dividing

To divide by a fraction, turn it upside down and multiply:

$$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\times \frac{d}{c}=\frac{ad}{bc}$$

The flipped fraction is called the reciprocal. The rule looks like a trick, but the question makes it obvious: 3 ÷ 1/2 asks how many halves fit inside 3, and the answer is 6. Dividing by a half doubles. Dividing by 1/4 quadruples. The reciprocal is doing exactly what it looks like.

Reducing to lowest terms

220/440 and 1/2 are the same number, and the second is easier to work with. To reduce a fraction, divide the top and the bottom by their greatest common factor. For 217/98 that factor is 7, which gives 31/14.

Nothing is lost in reducing. The value is identical, only the description is shorter. A fraction is in lowest terms when the numerator and denominator share no factor except 1, which is what the simplify calculator reports as the greatest common factor.

Turning a fraction into a decimal

Divide the top by the bottom. Some fractions stop and some never do, and there is a clean rule for telling them apart before you start. Reduce the fraction first, then look at the denominator. If its only prime factors are 2s and 5s, the decimal terminates. Anything else repeats forever.

Fraction	Denominator factors	Decimal
3/8	2 × 2 × 2	0.375, stops
7/20	2 × 2 × 5	0.35, stops
1/3	3	0.333… repeats
2/7	7	0.285714285714… repeats

The reason is that our decimal system is built on tens, and 10 is 2 × 5. A denominator made only of 2s and 5s can be scaled up to a power of ten, so it fits the notation. A 3 or a 7 never will.

The repeat in 2/7 has a block of six digits, 142857, and it runs forever. Any calculator that shows you 0.28571428571429 has rounded. The last digit is a lie of convenience, which matters if you are chaining calculations rather than reading one answer.

Turning a decimal into a fraction

Count the digits after the point. That count is the power of ten you need underneath. 1.375 has three decimal places, so it starts as 1375/1000, and reducing by 125 gives 11/8.

0.1234 has four places, so it begins as 1234/10000 and reduces by 2 to 617/5000. The method is mechanical, and the only judgement involved is spotting the greatest common factor at the end.

Repeating decimals need different handling. 0.333… is not 333/1000 or 3333/10000, it is exactly 1/3, and no finite decimal will ever land on it. The decimal-to-fraction calculator above reads the digits you type as an exact terminating value.

Why the bottom number cannot be zero

3/0 has no answer. Division asks what number multiplied by the denominator gives the numerator, and nothing multiplied by 0 gives 3. It is not enormous or infinite, it is undefined, which is why every calculator here refuses the input rather than showing a number.

Where fractions still turn up

Anything measured in inches is fractional. A tape measure divides the inch into halves, quarters, eighths and sixteenths, and drill bits and spanners are sized the same way, which is why 7/16 and 3/8 are sizes rather than calculations. Recipes scale by fractions, and doubling 2/3 of a cup is a fraction problem before it is a cooking one.

American share prices were quoted in fractions of a dollar for two centuries, in eighths and later sixteenths, until the exchanges finished moving to decimals on 9 April 2001. Fractions are also older than the notation we use for them: the Rhind Mathematical Papyrus, copied around 1550 BC, works almost entirely in unit fractions with 1 on top.

Reading your result

Take the simplified fraction as the answer. The mixed number is the same value in a form that is easier to picture, and the decimal is the same value again, rounded. If you are carrying the number into further work, use the fraction. If you are reading it off a page or telling someone, use whichever of the three lands fastest.