Question

Convert .454545454545(repeating decimal) to rational number in simplest form.

Answer 1

To convert the repeating decimal \( x = 0.454545454545... \) into a rational number, we can use the following method:

1. **Set up the equation**:
\[
x = 0.454545454545...
\]

2. **Multiply both sides by 100** (shifting the decimal point two places to the right):
\[
100x = 45.454545454545...
\]

3. **Now we have two equations**:
\[
x = 0.454545454545... \quad (1)
\]
\[
100x = 45.454545454545... \quad (2)
\]

4. **Subtract equation (1) from equation (2)**:
\[
100x - x = 45.454545454545... - 0.454545454545...
\]
This simplifies to:
\[
99x = 45
\]

5. **Solve for \( x \)**:
\[
x = \frac{45}{99}
\]

6. **Simplify the fraction**:
To simplify \( \frac{45}{99} \), we can find the greatest common divisor (GCD) of 45 and 99. The GCD is 9. Now we divide both the numerator and the denominator by 9:
\[
x = \frac{45 \div 9}{99 \div 9} = \frac{5}{11}
\]

Therefore, the repeating decimal \( 0.454545454545... \) can be expressed as the rational number in simplest form:
\[
\frac{5}{11}
\]