Question

which fraction is equivalent to `0.\overline{57}`?

Answer 1

To convert the repeating decimal \(0.\overline{57}\) into a fraction, we can use the following steps:

Let \(x = 0.\overline{57}\).

This means we can express \(x\) as:
\[
x = 0.57575757\ldots
\]

To eliminate the repeating part, we can multiply \(x\) by 100 because the repeating block has two digits:
\[
100x = 57.57575757\ldots
\]

Now, we can set up the equation:
\[
100x = 57 + x
\]

Next, we can solve for \(x\) by subtracting \(x\) from both sides:
\[
100x - x = 57
\]
\[
99x = 57
\]

Now, divide both sides by 99:
\[
x = \frac{57}{99}
\]

Next, we simplify the fraction \(\frac{57}{99}\). The greatest common divisor (GCD) of 57 and 99 is 9, so we divide both the numerator and denominator by 9:
\[
\frac{57 \div 9}{99 \div 9} = \frac{6}{11}
\]

Thus, the fraction that is equivalent to \(0.\overline{57}\) is:
\[
\boxed{\frac{6}{11}}
\]