Question

how to write a repeating decimal as a fraction

Answer 1

To convert a repeating decimal into a fraction, follow these steps. Let's use the example of the repeating decimal \( 0.666...\) (which repeats the digit 6).

1. **Set up the equation**: Let \( x = 0.666...\).

2. **Multiply by a power of 10**: Since the repeating part is one digit long, multiply both sides of the equation by 10:
\[
10x = 6.666...
\]

3. **Set up a second equation**: Now you have two equations:
\[
x = 0.666...
\]
\[
10x = 6.666...
\]

4. **Subtract the first equation from the second**:
\[
10x - x = 6.666... - 0.666...
\]
\[
9x = 6
\]

5. **Solve for \( x \)**:
\[
x = \frac{6}{9}
\]

6. **Simplify the fraction**: Divide both the numerator and the denominator by their greatest common divisor (3):
\[
x = \frac{2}{3}
\]

So, \( 0.666... \) as a fraction is \( \frac{2}{3} \).

### Another Example

Let's consider another repeating decimal, \( 0.2\overline{3} \) (which is \( 0.23333...\)).

1. **Set up the equation**: Let \( x = 0.23333...\).

2. **Identify the non-repeating and repeating parts**:
- The non-repeating part is 0.2.
- The repeating part is 0.03333...

3. **Multiply for the non-repeating part**: Since the non-repeating part is one digit long, multiply by 10:
\[
10x = 2.3333...
\]

4. **Multiply for the repeating part**: Multiply by 100 because the repeating part (3) has one digit:
\[
100x = 23.3333...
\]

5. **Subtract the two equations**:
\[
100x - 10x = 23.3333... - 2.3333...
\]
\[
90x = 21
\]

6. **Solve for \( x \)**:
\[
x = \frac{21}{90}
\]

7. **Simplify the fraction**: Divide both the numerator and the denominator by 3:
\[
x = \frac{7}{30}
\]

So, \( 0.2\overline{3} \) as a fraction is \( \frac{7}{30} \).

This method can be applied to any repeating decimal!