Question

How do you convert the repeating decimal 0.363636 to a rational number?

Answer 1

To convert the repeating decimal \( 0.363636\ldots \) (which we denote as \( x \)) to a rational number, we start by expressing the number as follows:

Let \( x = 0.363636\ldots \).

Next, we notice that the decimal repeats every two digits. To eliminate the repeating part, we can multiply \( x \) by 100 (since the repeating block has 2 digits):

\[
100x = 36.363636\ldots
\]

Now we have two equations:

1. \( x = 0.363636\ldots \)
2. \( 100x = 36.363636\ldots \)

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

\[
100x - x = 36.363636\ldots - 0.363636\ldots
\]

This simplifies to:

\[
99x = 36
\]

Now, to find \( x \), we divide both sides by 99:

\[
x = \frac{36}{99}
\]

We can simplify this fraction. The greatest common divisor (GCD) of 36 and 99 is 9. So we divide the numerator and denominator by 9:

\[
x = \frac{36 \div 9}{99 \div 9} = \frac{4}{11}
\]

Therefore, the repeating decimal \( 0.363636\ldots \) can be expressed as the rational number

\[
\boxed{\frac{4}{11}}.
\]