Question

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


57/9
57/10
57/99
57/100

Answer 1

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

1. Let \(x = 0.\overline{57}\).
2. Multiply both sides by 100 (because the repeating part has 2 digits):
\[
100x = 57.\overline{57}
\]
3. Now, subtract the original \(x\) from this equation:
\[
100x - x = 57.\overline{57} - 0.\overline{57}
\]
This simplifies to:
\[
99x = 57
\]
4. Now, solve for \(x\):
\[
x = \frac{57}{99}
\]

Now, we can reduce \(\frac{57}{99}\) to its simplest form. Both the numerator and the denominator can be divided by 9:

- \(57 \div 9 = 6.33\) (not applicable in this case)
- \(99 \div 9 = 11\)

Therefore, \( \frac{57}{99} \) simplifies to \( \frac{19}{33} \), but since we are only concerned with equivalency in the context of the options given:

Comparing with the options:
- \( \frac{57}{9} \)
- \( \frac{57}{10} \)
- \( \frac{57}{99} \) (this one matches with the derived fraction)
- \( \frac{57}{100} \)

The correct equivalent fraction for \(0.\overline{57}\) is \(\frac{57}{99}\).