Question

Convert the following repeating decimal to a fraction in simplest form. 0.30 (30 is repeated)

Answer 1

Let \( x = 0.303030...\).

To eliminate the repeating decimal, we can multiply \( x \) by 100 (since the repeating part "30" has two digits):

\[
100x = 30.303030...
\]

Now we can set up a system of equations:

1. \( x = 0.303030... \)
2. \( 100x = 30.303030... \)

Next, we subtract the first equation from the second:

\[
100x - x = 30.303030... - 0.303030...
\]

This simplifies to:

\[
99x = 30
\]

Now, solve for \( x \):

\[
x = \frac{30}{99}
\]

Next, we simplify the fraction \( \frac{30}{99} \). To do this, we find the greatest common divisor (GCD) of 30 and 99.

The prime factorization of 30 is \( 2 \times 3 \times 5 \) and for 99 it is \( 3^2 \times 11 \). The GCD is 3.

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

\[
\frac{30 \div 3}{99 \div 3} = \frac{10}{33}
\]

Thus, the repeating decimal \( 0.303030... \) can be expressed as the fraction \( \frac{10}{33} \).

In conclusion, the simplified form of the repeating decimal \( 0.30(30) \) is:

\[
\boxed{\frac{10}{33}}
\]