Number
Grade 7-8Recurring Decimals to Fractions
Every recurring decimal is secretly a fraction, and the algebraic trick for finding it is always the same: multiply by a power of 10 that lines up the repeating part, then subtract to make the repeating decimals cancel out. This lesson covers converting recurring decimals with one repeating digit, two or more repeating digits, and recurring decimals that have non-repeating digits before the repeating block.
What you need to know
- Every recurring decimal is a fraction, so it is a rational number.
- Let x be the decimal, then multiply by 10, 100 or 1000 to line up the repeating block.
- Subtract the two equations to remove the recurring part, then solve for x.
- One repeating digit means multiply by 10; two repeating digits means multiply by 100.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Converting a recurring decimal with one repeating digit
Let x equal the recurring decimal, then multiply both sides by 10 so the repeating part lines up exactly. Subtracting the original equation removes the recurring part entirely, leaving a simple equation to solve for x.
0.7 recurring: let x = 0.7̇, so 10x = 7.7̇. Subtracting gives 10x - x = 7.7̇ - 0.7̇, so 9x = 7, and x = 7/9.
Converting a recurring decimal with two or more repeating digits
When two digits repeat, multiply by 100 instead of 10, so the repeating block lines up before subtracting.
0.18 recurring (both digits repeating): let x = 0.18̇1̇8̇, so 100x = 18.18̇1̇8̇. Subtracting gives 99x = 18, so x = 18/99, which simplifies to 2/11.
Recurring decimals with non-repeating digits first
When some digits after the decimal point do not repeat, multiply by two different powers of 10, one that moves the non-repeating part in front of the decimal point, and one that also moves one full repeating block. Subtracting these two equations removes the recurring part.
0.16 recurring (only the 6 repeats): let x = 0.16̇, so 10x = 1.6̇ and 100x = 16.6̇. Subtracting 10x from 100x gives 90x = 15, so x = 15/90, which simplifies to 1/6.
Proving a recurring decimal equals a given fraction
"Prove that" questions expect the full algebraic method shown clearly, not just the final fraction, since the marks are for the method rather than for stating the already-given answer.
To prove 0.36 recurring equals 4/11: let x = 0.36̇3̇6̇, so 100x = 36.36̇3̇6̇. Subtracting gives 99x = 36, so x = 36/99, which simplifies to 4/11, as required.
Worked examples
Three exam-style questions, fully solved
Convert 0.7 recurring to a fraction in its simplest form.
Easy- 1.Let x = 0.7 recurring
- 2.Multiply by 10: 10x = 7.7 recurring
- 3.Subtract to remove the recurring part: 10x - x = 7.7 recurring - 0.7 recurring, giving 9x = 7
Answer: 7/9
Convert 0.16 recurring (only the 6 repeats) to a fraction in its simplest form.
Medium- 1.Let x = 0.16 recurring
- 2.Multiply by 10 to move the non-repeating digit: 10x = 1.6 recurring
- 3.Multiply by 100 to also move one repeating block: 100x = 16.6 recurring
- 4.Subtract: 100x - 10x = 16.6 recurring - 1.6 recurring, giving 90x = 15
Answer: 1/6
Prove that 0.36 recurring is equal to 4/11.
Hard- 1.Let x = 0.36 recurring, where both digits repeat
- 2.Multiply by 100: 100x = 36.36 recurring
- 3.Subtract: 100x - x = 36.36 recurring - 0.36 recurring, giving 99x = 36
- 4.Solve and simplify: x = 36/99 = 4/11, as required
Answer: 4/11
Practice
6 questions, marked instantly
Type an answer and check it. The worked solution appears once you have had a go. No login needed, and your progress saves in this browser.
Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
Write 0.7777… (the 7 repeats) as a fraction.
Write 0.4444… (the 4 repeats) as a fraction in its simplest form.
Write 0.272727… (the 27 repeats) as a fraction in its simplest form.
Write 0.1111… (the 1 repeats) as a fraction.
Write 0.363636… (the 36 repeats) as a fraction in its simplest form.
Write 0.6666… (the 6 repeats) as a fraction in its simplest form.
Avoid these
The mistakes students make most often
Multiplying by the wrong power of 10, for example multiplying by 10 when two digits repeat instead of 100.
Not accounting for non-repeating digits before the repeating block, and using a single multiplication step instead of two different powers of 10.
Losing track of what "x" represents when subtracting the two equations, and subtracting the wrong pair of values.
Leaving the final fraction unsimplified, instead of dividing the numerator and denominator by their highest common factor.
In a "prove that" question, writing only the final fraction without showing the full algebraic method, which loses the method marks even when the answer is correct.
FAQ
Questions students and parents ask
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