Convert Recurring Decimals to Fractions

Q: Why do you multiply by 10 or 100?

You multiply by a power of 10 to shift the digits so the repeating parts line up. Use 10 when a single digit repeats, and 100 when two digits repeat. Once the repeating parts align, subtracting the two equations removes them exactly.

Q: Why does subtracting the equations work?

Both equations end in the same never-ending repeating tail. When you subtract one equation from the other, those identical tails cancel out completely. That leaves a simple equation with whole numbers, which you can then solve and simplify to find x.

Q: Can every recurring decimal be written as a fraction?

Yes. Every recurring decimal can be written as an exact fraction. Whatever digits repeat, you set the decimal equal to x. You then build and subtract equations to remove the repeating part, and simplify the result to reach the fraction.

Key concept

Convert recurring decimals to fractions by letting x equal the decimal, then multiplying by a power of 10 so the repeating digits line up. Subtracting cancels them: if x = 0.333…, then 10x = 3.333…, so 10x − x = 3 and x = 1/3.

Convert Recurring Decimals to Fractions - introduction visual

Video Lesson

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$Convert Recurring Decimals to Fractions poster$

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Flashcards

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$Step-by-step guide converting 0.333 recurring decimal to fraction 1/3 by multiplying by 10, subtracting equations, and solving for x.$ $Converting the recurring decimal 0.4545... into a fraction, multiplying both sides by 100, subtracting the equations, and solving for x = 5/11.$ $Converting the recurring decimal 0.1666... into a fraction, showing multiplication by powers of 10, subtraction of equations, and solving to get 1/6.$

Converting Recurring Decimals to Fractions

Let x equal the recurring decimal, for example $x = 0.333$ ….
Multiply by 10 so the repeating digits line up, then subtract 10x − x.
Solve $9 x = 3$ , so $x = 1/3$ , meaning 0.333… $= 1/3$ .

Practice Questions

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Q1Easy

Convert $0. \overline{7}$ to a fraction.

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Interactive Activity

Learn how to convert recurring decimals into fractions

Write the Simplified Fraction

What is 0.•3 as a fraction?

Enter the simplified fraction

Convert

0.•3

to a fraction

Let x equal the decimal

Let x = 0.333...

Multiply both sides by 10

Click to reveal

Subtract to remove repeating part

• • •

So 9x = 3

• • •

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Students Also Ask

The questions students bump into most on this topic

Why do you let x equal the recurring decimal?

Letting x equal the recurring decimal turns it into a value you can work with using algebra. From that one equation you can build a second equation with the same repeating tail. Subtracting them removes the repeating part and lets you solve for x as a fraction.

Why do you multiply by 10 or 100?

You multiply by a power of 10 to shift the digits so the repeating parts line up. Use 10 when a single digit repeats, and 100 when two digits repeat. Once the repeating parts align, subtracting the two equations removes them exactly.

Why does subtracting the equations work?

Both equations end in the same never-ending repeating tail. When you subtract one equation from the other, those identical tails cancel out completely. That leaves a simple equation with whole numbers, which you can then solve and simplify to find x.

What if only some of the digits repeat?

Sometimes a digit does not repeat before the recurring part begins. In that case, you make more equations by multiplying by 10 and by 100. You then subtract the pair that lines up the repeating part, for example 100x and 10x, then solve as before.

Can every recurring decimal be written as a fraction?

Yes. Every recurring decimal can be written as an exact fraction. Whatever digits repeat, you set the decimal equal to x. You then build and subtract equations to remove the repeating part, and simplify the result to reach the fraction.

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