Question

Convert 0.72 (72 is under a repeating bar) to a rational number in simplest form.

Answer 1

Let \( x = 0.72\overline{72} \), meaning the decimal 0.727272... where "72" is repeating.

To eliminate the repeating decimal, we can multiply \( x \) by \( 100 \) (since the repeating part is 2 digits long):

\[
100x = 72.\overline{72}
\]

Now we have two equations:
1. \( x = 0.72\overline{72} \)
2. \( 100x = 72.\overline{72} \)

Next, we subtract the first equation from the second:

\[
100x - x = 72.\overline{72} - 0.72\overline{72}
\]

This simplifies to:

\[
99x = 72
\]

Now, we solve for \( x \):

\[
x = \frac{72}{99}
\]

Next, we simplify the fraction \( \frac{72}{99} \). We need to find the greatest common divisor (GCD) of 72 and 99.

The prime factorization of 72 is \( 2^3 \times 3^2 \) and for 99 it is \( 3^2 \times 11 \). The GCD is \( 3^2 = 9 \).

Now, we can simplify \( \frac{72}{99} \) by dividing both the numerator and denominator by their GCD:

\[
\frac{72 \div 9}{99 \div 9} = \frac{8}{11}
\]

Thus, the repeating decimal \( 0.72\overline{72} \) can be expressed as the rational number:

\[
\boxed{\frac{8}{11}}
\]