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?

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!