## Repeating Decimals

## Work Time

# Repeating Decimals

To indicate that a decimal repeats forever in a specific pattern, you write a bar over the repeating digits.

For example, $\frac{6}{11}=0.54545454545\dots $. You can write this decimal as $0.\overline{54}$. The use of a line to show the repeating digits in a decimal is called *bar notation*.

Write each of these fractions in decimal form using bar notation.

- $\frac{1}{3}$
- $\frac{2}{3}$
- $\frac{11}{12}$
- $\frac{3}{11}$

# Challenge Problem

It is easy to change a terminating decimal to a fraction; for example, $0.09=\frac{9}{100}$ and $3.2=\frac{32}{10}$.

Changing a repeating decimal to a fraction is trickier. The steps that follow describe a method for changing the repeating decimal $0.\overline{12}$ to a fraction.

- Write the decimal to show the repeating pattern: For $0.\overline{12}$, you write 0.1212121212…

- Let
*x*equal the repeating decimal:*x*= 0.1212121212…

- Multiply both sides of the preceding equation by whatever power of 10 (10, 100, 1000, and so on) moves “one set” of the repeating digits to the left of the decimal point. One set of repeating digits in 0.1212121212… is “12.” To move this set to the left of the decimal point, you need to multiply
*x*by 100: 100*x*= 12.1212121212… Subtract

*x*from the equation in the previous step:100 *x*= 12.1212121212… − *x*= 0.1212121212… 99 *x*= 12 - Solve $x=\frac{12}{99}=\frac{4}{33}$

** • Use this method to change $0.\overline{7}$ to a fraction.**