Question

The fraction that is equivalent to 0.45(reapeating deciaml)_

Answer 1

To convert the repeating decimal \(0.45\overline{45}\) into a fraction, we can use the following method:

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

To eliminate the repeating part, we can multiply both sides of the equation by 100 (because the repeating part has two digits):

\[
100x = 45.454545...
\]

Now we have two equations:

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

Next, we can subtract the first equation from the second:

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

This simplifies to:

\[
99x = 45
\]

Now, we solve for \(x\):

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

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 9:

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

Therefore, the fraction that is equivalent to \(0.45\overline{45}\) is:

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