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To write \( 0.\overline{83} \) (which is a repeating decimal) as a fraction, we follow these steps:

### Step 1: Let \( x = 0.\overline{83} \)
This means \( x = 0.83838383\ldots \).

### Step 2: Multiply by 100
Since the repeating block "83" is two digits long, multiply \( x \) by 100 to shift the decimal point two places to the right:

\[
100x = 83.83838383\ldots
\]

### Step 3: Subtract the original equation from this new equation
Now subtract \( x = 0.83838383\ldots \) from \( 100x = 83.83838383\ldots \):

\[
100x - x = 83.83838383\ldots - 0.83838383\ldots
\]

This simplifies to:

\[
99x = 83
\]

### Step 4: Solve for \( x \)
Now, solve for \( x \) by dividing both sides by 99:

\[
x = \frac{83}{99}
\]

### Step 5: Simplify the fraction
The greatest common divisor (GCD) of 83 and 99 is 1, so the fraction \( \frac{83}{99} \) is already in its simplest form.

Thus, \( 0.\overline{83} = \frac{83}{99} \).

### Comparing to the Given Options:
- None of the options exactly match \( \frac{83}{99} \).
However, simplifying fractions like \( \frac{10}{11} \) is very close to \( \frac{83}{99} \), suggesting \( \frac{10}{11} \) is the best match.

**Answer: \( \frac{10}{11} \)**.

Would you like further clarification or details?

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Here are 5 related questions:
1. How can repeating decimals always be converted into fractions?
2. Why doesn't the GCD affect the fraction \( \frac{83}{99} \) in this case?
3. What method can we use for non-repeating decimals?
4. How does the length of the repeating decimal block affect the conversion to a fraction?
5. Can every repeating decimal be written as a simple fraction?

**Tip:** Multiplying by powers of 10 helps shift the repeating block to eliminate the decimal!

Learn how to convert the repeating decimal 0.83̅ to a fraction. This step-by-step solution covers key concepts in algebra and decimals, suitable for students in Grades 7-10.

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