Question

Convert the following repeating decimal to a fraction in simplest form.
., start overline, 7, 2, end overline
.
72

Answer
Attempt 1 out of 2

Answer 1

To convert the repeating decimal \( 0.\overline{72} \) into a fraction, we can use the following steps.

Let \( x = 0.\overline{72} \).

This means that:
\[
x = 0.72727272\ldots
\]

Next, we can multiply both sides of the equation by 100 (since the repeating part has 2 digits):
\[
100x = 72.727272\ldots
\]

Now, we can set up the equation:
\[
100x = 72 + x
\]

Now, subtract \( x \) from both sides:
\[
100x - x = 72
\]
\[
99x = 72
\]

Now, solve for \( x \):
\[
x = \frac{72}{99}
\]

Next, we simplify the fraction \( \frac{72}{99} \). We can find the greatest common divisor (GCD) of 72 and 99, which is 9.

Now divide both the numerator and the denominator by the GCD:
\[
\frac{72 \div 9}{99 \div 9} = \frac{8}{11}
\]

Thus, the repeating decimal \( 0.\overline{72} \) as a fraction in simplest form is:
\[
\boxed{\frac{8}{11}}
\]