Question

Convert 0.72¯¯¯¯¯ to a rational number in simplest form.(1 point) please

Answer 1

To convert the repeating decimal \( 0.72\overline{72} \) to a rational number, we can follow these steps:

Let \( x = 0.72\overline{72} \).

To eliminate the repeating part, we can multiply \( x \) by a power of 10. Since the repeat is 2 digits long, we'll multiply by \( 100 \):

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

Now, we have:
\[
x = 0.72\overline{72}
\]
\[
100x = 72.\overline{72}
\]

Next, we subtract the first equation from the second:

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

This simplifies to:

\[
99x = 72
\]

Now, divide both sides by \( 99 \):

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

Next, we simplify the fraction. The greatest common divisor (GCD) of 72 and 99 is 9.

Dividing the numerator and the denominator by 9 gives:

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

Thus, the repeating decimal \( 0.72\overline{72} \) converted to a rational number in simplest form is:

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