How to convert recurring decimals to fractions
A recurring decimal, also known as a repeating decimal, is a number containing an infinitely repeating digit – or series of digits – occurring after the decimal point. For example 0.3333 recurring, or 1.7454545 recurring.
The recurring digit or digits are typically identified by a dot placed above them, so 0.3 with a dot above the 3, or 1.745 with dots above both the 4 and 5. Where there is a long series of repeating digits, dots appear above the first and last digits of the recurring sequence. For example, for 0.385385 recurring, you would see dots above both the 3 and the 5.
Since recurring decimals are rational numbers, they can always be expressed as fractions. Questions that require you to convert a recurring decimal to a fraction often crop up in numerical reasoning tests, so understanding the process is key.
Decimals and fractions are essentially two different ways of representing the same numerical value. A key mathematical skill is knowing how to convert fractions to decimals and decimals to fractions.
How do I convert recurring decimals to fractions? We’ll walk through this step by step below.
Step 1: Write out the equation
To convert a recurring decimal to a fraction, start by writing out the equation where (the fraction we are trying to find) is equal to the given number. Use a few repeats of the recurring decimal here.
For example, if we’re asked to convert 0.6 recurring to a fraction, we would start out with:
𝓍 = 0.6666…
Step 2: Cancel out the recurring digits
To do this, we need a second equation with the same recurring digits after the decimal point. In this case, if we multiply both sides of 𝓍 = 0.6666… by 10, we get 10𝓍 = 6.6666…
As our recurring digits are the same in both equations, we can subtract the lesser from the greater to cancel them out:
10𝓍 - 𝓍 = 6.666… - 0.6666…
This gives us 9𝓍 = 6
Step 3: Solve for 𝒳
We now take 9𝓍 = 6 and divide both sides by 9 to find our fraction:
𝓍 = 6/9
Step 4: Simplify the fraction
When converting a recurring decimal to a fraction, we need to find its lowest form. In this case, both 6 and 9 are divisible by 3, so we complete our conversion by stating:
0.6 recurring = 2/3
The above is a basic example of how to convert a recurring decimal into a fraction. However, in most cases, step two is more complex.
As an example, let’s take 0.03666… as our recurring decimal, so 𝓍 = 0.03666…
If we multiply both sides by 10 we get 10𝓍 = 0.3666… Here, the recurring digits do not match our first equation, so can’t be cancelled out.
To resolve this, we must multiply 𝓍 = 0.03666…by 100 and also by 1000 to give us two equations with matching recurring digits:
100𝓍 = 3.666…
1000𝓍 = 36.666…
We would then move on to step three, subtracting the lesser from the greater:
1000𝓍 - 100𝓍 = 36.666… - 3.666…
900𝓍 = 33
Now we solve for 𝓍 and simplify the fraction:
𝓍 = 33/900
0.03666… = 11/300
Example question 1
Express 0.84 recurring as a fraction in its lowest form.
First, write out the equation as 𝓍 = 0.8484…
Find multiples of 𝓍 that result in two equations with the same recurring digits after the decimal point, in this case:
100𝓍 = 84.8484…
10000𝓍 = 8484.8484…
Subtract and solve for 𝓍:
10000𝓍 - 100𝓍 = 8484.8484… - 84.8484…
9900𝓍 = 8400
𝓍 = 8400/9900
Simplify:
0.84 recurring = 28/33
Example question 2
Express 0.0237237 recurring as a fraction in its lowest form.
Write out the equation as 𝓍 = 0.0237237…
Find multiples of 𝓍 that result in two equations with the same recurring digits after the decimal point, in this case:
10𝓍 = 0.237237…
10000𝓍 = 237.237237…
Subtract and solve for 𝓍:
10000𝓍 - 10𝓍 = 237.237237… - 0.237237…
9990𝓍 = 237
𝓍 = 237/9990
Simplify:
0.0237237 recurring = 79/3330
Example question 3
Express 7.322 recurring as a fraction in its lowest form.
Write out the equation as 𝓍 = 7.3222…
Find multiples of 𝓍 that result in two equations with the same recurring digits after the decimal point, in this case:
10𝓍 = 73.222…
100𝓍 = 732.222…
Subtract and solve for 𝓍:
100𝓍 - 10𝓍 = 732.222… - 73.222…
90𝓍 = 659
𝓍 = 659/90
In this instance we’ve been left with an improper fraction where the numerator is greater than the denominator, so we need to simplify this to a mixed fraction:
7.322 recurring = 7 and 29/90
Tips for converting recurring decimals to fractions
1) Count the number of times you move the decimal point
If you’re struggling to spot multiples that allow you to cancel out the recurring digits, write out the number and insert additional decimal points that leave the recurring digit to the right.
For example, if you’re converting 0.23434, your recurring digits are 34, so write out 0.2.34.34.
Now count the moves from the first decimal point to the second, and the first to the third. In this case the first move is 1, so add a zero to 1, making your first multiple 10. The second move is 3, so add 3 zeros to 1, making your second multiple 1000.
You now have your two equations of 10𝓍 = 2.3434… and 1000𝓍 = 234.3434…
2) Revise your times tables
In a numerical reasoning test, time is of the essence, and since most recurring decimal conversions will require you to simply the fraction, you need to be able to do this at speed. Knowing your times tables will help you quickly list common factors of the numerator and denominator.
Once you’ve listed these out, you can identify the greatest common factor to calculate the fraction’s lowest form.
3) Practice simplifying improper fractions
Some conversions will leave you with a numerator greater than the denominator, or an improper fraction. To simplify this, you need to turn it into a mixed fraction.
Do this by calculating how many times the denominator fits into the numerator as a whole – this is the whole number of your mixed fraction. Then take the remainder and place it over the denominator to give you your fractional part.
4) Learn to work in reverse
Your numerical reasoning test may throw up multiple questions on conversions, so learn to work in reverse and work out recurring decimals. You should also practice converting both fractions and decimals into percentages, and vice versa.