convert . 454545454545 repeating decimal to a rational number in simplest form

Answer 1

To convert the repeating decimal \(0.454545454545...\) (denoted as \(x\)) to a fraction, we can use the following method:

1. Let \(x = 0.454545...\).

2. Multiply both sides of the equation by 100 (since the repeating part has 2 digits): \[ 100x = 45.454545... \]

3. Now, we have two equations: \[ x = 0.454545... \] \[ 100x = 45.454545... \]

4. Subtract the first equation from the second: \[ 100x - x = 45.454545... - 0.454545... \] \[ 99x = 45 \]

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

6. Now, simplify the fraction: \[ \frac{45}{99} = \frac{5}{11} \] (Divide the numerator and denominator by their greatest common divisor, which is 9.)

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