Simplify a fraction

What this works out

Two fractions can look completely different and be the same number. Putting one in its lowest terms is what makes two of them comparable, and it is the step that turns an answer into a mark in an exercise book rather than an argument about whether 22/51 and 462/1071 are the same thing.

This tool does that, and then gives the three other ways the same value gets written: the mixed number, the decimal, and the percentage. The decimal is the interesting one, because most fractions do not have a finite decimal at all, and a calculator that hands back nine digits and stops has quietly changed the number.

The method

The two boxes are read as the decimals they were typed as, not as the binary values a browser holds them in. Both are then multiplied by the same power of ten until they are whole numbers. That step is allowed because multiplying the top and the bottom by the same figure never changes what a fraction is worth; it is also why the power of ten has to come from the longer of the two inputs, since scaling only far enough for one of them leaves the other fractional.

From there everything is whole-number arithmetic. Euclid’s algorithm finds the greatest common divisor: divide the larger number by the smaller, then divide the smaller by the remainder, and repeat until the remainder is zero. The divisor used on that last line is the greatest common divisor, and dividing both halves by it is the whole of simplification. The steps are shown in a table below the result, because that table is the proof of the answer rather than a decoration.

The decimal is produced by long division rather than by dividing two numbers and seeing what comes out. Each step multiplies the remainder by ten and divides again, and every remainder seen is recorded. There are only as many possible remainders as the denominator, so the sequence must either reach zero or repeat one it has already had — and the first repeat is exactly where the recurring block begins. That is why the brackets can be trusted: they are not pattern matching on the digits, they are the point at which the division provably started going round.

Before you read on

Euclid's algorithm on 1071 and 462 ends with the line 147 = 7 x 21 + 0. What is the greatest common divisor?

• A7 — the number it divided by.

  That is the quotient on that line. The algorithm never uses the quotients for anything.

• B21.

  Yes: the divisor on the line where the remainder reached zero.

• C147 — the last number that went in.

  That is the dividend. 147 does not divide 462, so it cannot be a common divisor of anything here.

21 — the divisor on the line where the remainder reached zero. Dividing both halves by it is the whole of simplification: 1071 / 21 = 51 and 462 / 21 = 22, and 51/22 cannot be reduced further because 21 was the greatest such number and not merely a convenient one. The algorithm is about 2,300 years old and is still the fastest way to do this; the table under the result prints every line of it, because that table is the proof of the answer rather than a decoration.

1. 1071 = 2 x 462 + 147

   Divide the larger by the smaller and keep the remainder. Anything dividing both 1071 and 462 must also divide that 147.

2.  462 = 3 x 147 +  21

   So the problem is now the smaller pair, 462 and 147 — the same question about smaller numbers.

3.  147 = 7 x  21 +   0

   The remainder is zero, so stop. 21 divides 147 exactly, and by the line above it therefore divides everything that came before.

4. 1071 / 21 = 51,  462 / 21 = 22

   Divide both halves by it. 51/22 is in its lowest terms.

A worked example

Simplify 1071/462, the pair Euclid’s algorithm is usually taught with.

In its lowest terms	51/22
As a mixed number	2 7/22
As a decimal	2.3(18)
Cancelled by	21
Rounded decimal	2.31818182
As a percentage	231.8182%

The working:

1071 = 2 x 462 + 147
 462 = 3 x 147 +  21
 147 = 7 x  21 +   0     <- last divisor, so gcd = 21

1071 / 21 = 51
 462 / 21 = 22

51 / 22 = 2 remainder 7  ->  2 and 7/22

long division of 7 by 22, remainders in the margin:
  0.3   remainder 4
  0.31  remainder 18
  0.318 remainder 4      <- 4 has been seen, so 18 repeats

So the decimal is 2.3(18), and 2.3181818… is exact rather than rounded. These are the same numbers asserted in this tool’s test file, so if the formula ever changes without this page changing with it, the build fails.

What it does not do

It simplifies one fraction at a time, and it does not add, subtract, multiply or divide two of them. It cannot take a recurring decimal as input — there is no way to type “nought point three recurring” into a number box, so 0.333 is read as exactly that and reduces to 333/1000. It does not find a simpler fraction that is merely close to the value entered, which is a different job from simplification and gives a different answer. And where a whole number grows past what a double can count in ones, the numeric boxes stand down and show a dash; the fraction, the mixed number and the decimal are text and stay exact.

A fraction is also a proportion. Where the question underneath is how much larger one number is than another, percentage increase gives the same relationship in the form most people expect to read it.