Question

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

Answer 1

Let \( x = 0.454545454545 \ldots \).

To eliminate the repeating decimal, we can multiply both sides of the equation by 100, since the repeating part (45) has a length of 2 digits:

\[
100x = 45.454545454545 \ldots
\]

Now we have two equations:

1. \( x = 0.454545454545 \ldots \)
2. \( 100x = 45.454545454545 \ldots \)

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

\[
100x - x = 45.454545454545 \ldots - 0.454545454545 \ldots
\]

This simplifies to:

\[
99x = 45
\]

Now, we can solve for \( x \):

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

Next, we simplify the fraction \( \frac{45}{99} \). We can find the greatest common divisor (GCD) of 45 and 99, which is 9.

Now we divide both the numerator and the denominator by their GCD:

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

Thus, the repeating decimal \( 0.454545454545 \ldots \) can be expressed as the rational number in simplest form:

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