freeCodeCamp/guide/english/mathematics/convert-decimals-to-fractions/index.md

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Convert Decimals to Fractions

There are two approaches to converting rational numbers from their decimal representation to a fraction, one for finite decimals and another approach for infinite decimal expansions such as 1/3 = 0.333.... We will start with finite decimal numbers and make our way to any decimal numbers, including many examples along the way to help make each step of the process clear.

Convert Finite Decimals to Fractions

Suppose we have a finite decimal number x that we wish to convert to a fraction. If the decimal has n digits after the decimal point, multiply and divide by 10^n

x = (x*10^n)/10^n

The numerator is an integer, as multiplying x by 10^n moves the decimal point to the right n spaces -- precisely the number of spaces we have -- and the denominator 10^n is an integer, so this is a fraction representing our number. If possible, it is best to simplify common factors from the numerator and denominator.

Example 1

Convert 0.25 to a fraction.

As x = 0.25 has n = 2 digits after the decimal point, we multiply by 10^2 = 100 to get

0.25 = (0.25*100)/100 = 25/100 = 5/20 = 1/4

and any of the three fractions on the right is a fractional representation of 0.25.

Example 2

Convert 0.002 to a fraction.

As x = 0.002 has n = 3 digits after the decimal point, we multiply by 10^3 = 1000 and have

0.002 = (0.002*1000)/1000 = 2/1000 = 1/500.

Example 3

Convert 0.103 to a fraction.

As x = 0.103 has n = 3 digits after the decimal point, we multiply by 1000 again,

0.103 = (0.103*1000)/1000 = 103/1000.

Convert Infinite Decimals to Fractions

If an infinite decimal expansion corresponds to a rational number, it must repeat after a certain point, for example 1/3 = 0.333... with 3s repeating, and 1/10 = 0.090909... with the 09 pattern repeating. This repeating pattern is what we use to convert the decimal representation to a fraction.

Suppose we have a rational number y whose decimal consists of a repeating pattern p which is m digits.

Example 4

Convert y = 0.090909... to a fraction.

We have a repeating pattern p = 09 of m = 2 digits. So we can write

y = p/(10^2) + p/[(10^2)^2] + p/[(10^2)^4] + p/[(10^2)^6] + ...

which is a geometric series with initial term a = p/100 = 9/100 and common ratio r = 1/100. Thus, we have

y = a/(1 - r) = (9/100)/(1 - 1/100) = (9/100)/(99/100) = 9/99 = 1/11

as expected.

First General Method

In general, our decimal y with repeating pattern p of m digits can be written as the geometric series

y = p/(10^m) + p/[(10^m)^2] + p/[(10^m)^3] + p/[(10^m)^4] + ... 
  = [p/(10^m)]/[1 - 1/(10^m)]
  = [p/(10^m)]/[(10^m - 1)/(10^m)]
  = p/(10^m - 1)

Just as with finite decimal numbers there is a quick and easy fraction that represents the number; it can be described using only the repeating pattern in the decimal. However, with infinite decimal expansions you can have examples where it is not just a repeating pattern, for example 1/110 = 0.0090909... has repeating pattern 09 after an initial 0, and 203/240 = 0.8458333... with repeating 3s eventually, but the initial 8458 as well.

Here we see that we have a finite start of the decimal that is not part of the pattern that repeats forever, and then an infinite decimal expansion afterwards, so we need only combine both of the methods seen above!

Example 5

Convert 0.8458333... to a fraction.

We have

0.8458333... = 0.8458 + 0.0000333...

and so with our first method we know that

0.8458 = 8458/10000 = 4229/5000

while the second term is four 0s followed by a repeating pattern where we can use our general method above. To deal with 0s we use our first trick, multiply and divide by 10^4 as there are four of them, to get the repeating pattern to start at the decimal point. Namely, we have

0.0000333... = (0.0000333...)*10000/10000 = (0.333...)/10000

and

0.333... = 3/10 + 3/100 + 3/1000 + ...
         = 3/9 = 1/3

using our general formula above. Hence 0.0000333... = (1/3)/10000 = 1/30000 and putting everything together,

0.8458... = 4229/5000 + 1/30000 = 25375/30000 = 203/240

as expected.

Generial Method

So let's put everything together into one process that works for all decimals, where we have a formula that tells us exactly what the fraction corresponding to a decimal number is.

1. Separate your decimal number into it's finite non-repeating part, and the pattern that repeats forever (for finite deicmals the repeating part is 0s).
2. If the initial, non-repeating part has n digits after the decimal, then the repeating pattern starts after n digits, so if we call our non-repeating part x and our repeating pattern p which is m digits in length, our decimal number is equal to the fraction.

(x*10^n)/10^n + [p/(10^m - 1)]/10^n = [(x*10^n)*(10^m - 1) + p]/[(10^n)*(10^m - 1)],

the first term just the finite conversion method above, the second term the general term from above (adjusted for the initial n 0s) and the right hand side just finding a common denominator and simplifying the left hand side.

Let's see one more example using this last formula.

Example 6

Convert 1.35285714285714... to a fraction.

The non-repeating part is the initial 1.35 while the repeating pattern is 285714, so for our formula above we have p = 285714, x = 1.35, n = 2 and m = 6. Pluggin these in gives

1.35285714285714... = [(1.35)*10^2(10^6 - 1) + 285714]/[10^2(10^6 - 1)]
                    = [135*999999 + 285714]/99999900
                    = 135285579/99999900
                    = 947/700.