Question

Convert the following repeating decimal to a fraction in simplest form.
.75 (5 is repeating

Answer 1

Let \( x = 0.755555\ldots \), where the digit \( 5 \) is repeating.

To convert this repeating decimal to a fraction, we can follow these steps:

1. **Multiply by a power of 10** to shift the decimal point so that the repeating part is aligned. Since the repeating portion (5) has one digit, we multiply by 10:
\[
10x = 7.55555\ldots
\]

2. **Multiply by another power of 10** to shift the decimal so that both \( x \) and \( 10x \) have the same repeating digits. Here, we multiply by 10 again:
\[
100x = 75.55555\ldots
\]

3. **Set up the equations**:
\[
10x = 7.55555\ldots \quad \text{(1)}
\]
\[
100x = 75.55555\ldots \quad \text{(2)}
\]

4. **Subtract equation (1) from equation (2)**:
\[
100x - 10x = 75.55555\ldots - 7.55555\ldots
\]
\[
90x = 68
\]

5. **Solve for \( x \)**:
\[
x = \frac{68}{90}
\]

6. **Simplify the fraction**:
To simplify \( \frac{68}{90} \), find the greatest common divisor (GCD) of 68 and 90. The GCD is 2.
\[
\frac{68 \div 2}{90 \div 2} = \frac{34}{45}
\]

Thus, the repeating decimal \( 0.755555\ldots \) as a fraction in simplest form is:
\[
\boxed{\frac{34}{45}}
\]