Question

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

Answer 1

Let \( x = 0.4545454545\ldots \).

To convert this repeating decimal to a fraction, we first observe the repeated part, which is "45". This consists of two digits.

To eliminate the repeating decimal, we can multiply \( x \) by 100 (since there are two digits that repeat):

\[
100x = 45.4545454545\ldots
\]

Now, we can set up an equation using the original \( x \):

\[
x = 0.4545454545\ldots
\]
\[
100x = 45.4545454545\ldots
\]

Next, we subtract the first equation from the second:

\[
100x - x = 45.4545454545\ldots - 0.4545454545\ldots
\]

This simplifies to:

\[
99x = 45
\]

Now, solve for \( x \):

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

Next, we simplify the fraction \( \frac{45}{99} \). The greatest common divisor (GCD) of 45 and 99 can be found. The prime factorization of both numbers is:

- \( 45 = 3^2 \times 5 \)
- \( 99 = 3^2 \times 11 \)

The GCD is \( 3^2 = 9 \).

Now, we divide both the numerator and the denominator by their GCD (9):

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

Thus, the decimal \( 0.4545454545\ldots \) can be expressed as the fraction:

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