Question

Convert 0.72¯¯¯¯¯
0.
72
¯
to a rational number in simplest form.(1 point)

Answer 1

To convert the repeating decimal \( 0.72\overline{72} \) to a rational number, we will use a method that involves setting up an equation.

Let \( x = 0.727272\ldots \).

Now, to eliminate the repeating part, we multiply \( x \) by 100 (since the repeating part has 2 digits):

\[
100x = 72.727272\ldots
\]

Next, we can subtract the original \( x \) from this equation:

\[
100x - x = 72.727272\ldots - 0.727272\ldots
\]
\[
99x = 72
\]

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

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

Next, we need to simplify \( \frac{72}{99} \).

We can find the greatest common divisor (GCD) of 72 and 99. The prime factorizations are as follows:

- \( 72 = 2^3 \times 3^2 \)
- \( 99 = 3^2 \times 11 \)

The common factor is \( 3^2 = 9 \).

Now we divide both the numerator and the denominator by their GCD, which is 9:

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

Thus, the simplifying of \( 0.72\overline{72} \) results in:

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