4.5 Recurring decimals — Learn

Prerequisites: multiplication and place value.

Objectives: Convert recurring decimals, including those with a non-recurring part, into fractions. This content is Higher only.

Learn the method

Set the recurring decimal equal to $x$. Multiply by a power of $10$ that moves one repeating block past the decimal point. Subtract the equations to remove the recurring tail. A dot above a digit marks the repeating block.

Worked examples

Example 1. Convert $0.\dot{6}$ to a fraction.

Let $x=0.666\ldots$.

$10x=6.666\ldots$.

Subtract the first equation from the second.

$9x=6$.

Divide by $9$.

$x=\frac{6}{9}=\frac{2}{3}$.

Example 2. Convert $0.1\dot{2}$ to a fraction.

Let $x=0.1222\ldots$.

$10x=1.222\ldots$ and $100x=12.222\ldots$.

Subtract to remove the recurring part.

$90x=11$, so $x=\frac{11}{90}$.

Watch out. For $0.1\dot{2}$, only the $2$ repeats. The number is $0.1222\ldots$, not $0.121212\ldots$. Check by expanding the decimal before setting up the equations.

Progress check

The keyboard discount is $\frac14$. As a decimal this is $0.25$, so it is not recurring. Convert $0.\dot{2}$ to a fraction: $x=0.222\ldots$, so $10x-x=2$, $9x=2$, and $x=\frac29$.

You should now be able to

• convert a recurring decimal to a fraction;
• convert a recurring decimal with a non-recurring part to a fraction;
• use subtraction to remove a repeating tail.

Covers: [N10].