Convert the following repeating decimal to a fraction in simplest form.

., start overline, 7, 2, end overline
.
72

Answer
Attempt 1 out of 2

1 answer

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