Convert the following repeating decimal to a fraction in simplest form.
point, 4, start overline, 6, end overline
.4 
6
 
Answer
Attempt 1 out of 2

To convert the repeating decimal \( 0.\overline{46} \) to a fraction, we can follow these steps:

### Step 1: Set the repeating decimal equal to a variable.
Let \( x = 0.\overline{46} \).

This means:
\[
x = 0.46464646\ldots
\]

### Step 2: Eliminate the repeating part by multiplying by an appropriate power of 10.
Since the repeating block "46" has two digits, multiply both sides of the equation by 100:
\[
100x = 46.464646\ldots
\]

### Step 3: Subtract the original equation from the new equation.
Now subtract the first equation \( x = 0.464646\ldots \) from \( 100x = 46.464646\ldots \):
\[
100x - x = 46.464646\ldots - 0.464646\ldots
\]
\[
99x = 46
\]

### Step 4: Solve for \( x \).
\[
x = \frac{46}{99}
\]

### Step 5: Simplify the fraction.
The greatest common divisor (GCD) of 46 and 99 is 1, so the fraction is already in its simplest form.

Thus, the fraction representation of \( 0.\overline{46} \) is:
\[
\boxed{\frac{46}{99}}
\]

Would you like a further breakdown or additional examples?

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Here are 5 questions to expand on this:
1. How would you convert \( 0.\overline{123} \) to a fraction?
2. What is the repeating decimal for \( \frac{7}{11} \)?
3. Can you explain why multiplying by powers of 10 helps remove the repeating part of a decimal?
4. What is the fraction for \( 0.\overline{7} \)?
5. How can you convert a non-repeating decimal like \( 0.25 \) to a fraction?

**Tip**: Always check if the resulting fraction can be simplified by finding the greatest common divisor (GCD) of the numerator and denominator.

Convert the following repeating decimal to a fraction in simplest form. .4̅6̅

Learn how to convert the repeating decimal .46̅ into a fraction. This detailed step-by-step guide covers the algebraic method of eliminating the repeating part and solving for the fraction. Suitable for students in Grades 8-10.

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