Question

Convert 0.45--------- to a fraction.

Answer 1

To convert the repeating decimal \(0.45\overline{45}\) (where "45" repeats indefinitely) into a fraction, we can follow these steps:

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

To eliminate the repeating part, we can multiply \(x\) by 100 (since "45" has 2 digits):

\[
100x = 45.454545...
\]

Now, we have two equations:

1. \(x = 0.454545...\)
2. \(100x = 45.454545...\)

Next, we subtract equation 1 from equation 2:

\[
100x - x = 45.454545... - 0.454545...
\]

This simplifies to:

\[
99x = 45
\]

Now, solve for \(x\):

\[
x = \frac{45}{99}
\]

To simplify \(\frac{45}{99}\), we can divide the numerator and the denominator by their greatest common divisor (GCD), which is 9:

\[
\frac{45 \div 9}{99 \div 9} = \frac{5}{11}
\]

Thus, the repeating decimal \(0.45\overline{45}\) is equivalent to the fraction:

\[
\frac{5}{11}
\]