https://file.aiquickdraw.com/mathbot/file/173043244359176pgijls.jpg

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.

Learn how to convert the repeating decimal 0.37̅ to a simplified fraction with step-by-step instructions. Understand key concepts like converting repeating decimals and simplifying fractions. Suitable for students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation