https://file.aiquickdraw.com/mathbot/file/1730304942899ov5pngin.jpg

The task here is to convert the repeating decimal \(0.\overline{345}\) into a fraction by setting up equations and simplifying them. Let's go through the solution step by step.

### Step-by-Step Solution

1. **Define the variable:**
   Let:
   \[
   x = 0.\overline{345}
   \]

2. **Multiply by powers of 10 to shift the decimal:**
   To eliminate the repeating decimal, we will multiply by powers of 10 that match the length of the repeating segment (3 digits in this case, so by \(10^3 = 1000\)).

   - \(10x = 3.453\) (shift once)
   - \(100x = 34.534\) (shift twice)
   - \(1000x = 345.345\) (shift three times)

3. **Set up the subtraction equation:**
   Now subtract the equation for \(x\) from the equation for \(1000x\) to isolate the repeating part:
   \[
   1000x - x = 345.345 - 0.345
   \]
   This simplifies to:
   \[
   999x = 345
   \]

4. **Solve for \(x\):**
   Divide both sides by 999:
   \[
   x = \frac{345}{999}
   \]

5. **Simplify the fraction:**
   Now, reduce \(\frac{345}{999}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 3:
   \[
   x = \frac{345 \div 3}{999 \div 3} = \frac{115}{333}
   \]

So, the decimal \(0.\overline{345}\) as a fraction is:
\[
x = \frac{115}{333}
\]

Would you like further details on any of these steps?

### Related Questions

1. How can I convert a different repeating decimal, such as \(0.\overline{456}\), into a fraction?
2. What are the general steps to convert any repeating decimal to a fraction?
3. How would the solution change if the decimal were non-repeating?
4. Why do we subtract equations in steps like this when converting repeating decimals?
5. How can we verify that \(\frac{115}{333}\) is the simplest form of this fraction?

### Tip
To convert a repeating decimal to a fraction, remember to use powers of 10 that match the length of the repeating sequence.

Use equations to write 0.345 as a fraction.

Learn how to convert the repeating decimal 0.345 to a fraction with a step-by-step algebraic approach. This tutorial covers setting up equations, shifting decimals, and simplifying fractions. Ideal for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation