To rewrite $0.\overline{37}$ as a simplified fraction, let’s go through the steps to convert this repeating decimal to a fraction.

Step-by-Step Solution

1. Let $x = 0.\overline{37}$. This means $x = 0.37373737\ldots$.

2. Multiply $x$ by 100 (since two digits are repeating) to shift the decimal point two places to the right: $100x = 37.373737\ldots$

3. Now we have: $x = 0.373737\ldots$ $100x = 37.373737\ldots$

4. Subtract the first equation from the second to eliminate the repeating decimal part: $100x - x = 37.373737\ldots - 0.373737\ldots$ This simplifies to: $99x = 37$

5. Solve for $x$: $x = \frac{37}{99}$

6. Check for simplification: The fraction $\frac{37}{99}$ is already in its simplest form since 37 and 99 have no common factors other than 1.

Final Answer

$0.\overline{37} = \frac{37}{99}$

Would you like more details on any part of this process?

Related Questions

1. How can we convert a repeating decimal with three repeating digits into a fraction?
2. What is the general method for converting a repeating decimal to a fraction?
3. How can we convert $0.\overline{123}$ into a fraction?
4. What is the difference between terminating and repeating decimals in terms of fractions?
5. How can a repeating decimal that starts after a few digits (e.g., 0.123\overline{4}) be converted into a fraction?

Tip

For any repeating decimal $0.\overline{ab}$ (where $a$ and $b$ are digits), the simplified fraction can often be obtained by using $\frac{ab}{99}$, provided $ab$ and 99 are coprime.