Question

convert the following repeating decimal to a fraction in simplest form.
.\overline{82}

Explanation:

Step1: Let \( x = 0.\overline{82} \)

Let \( x \) represent the repeating decimal. So \( x = 0.828282\ldots \)

Step2: Multiply \( x \) by 100

Since the repeating part has 2 digits, we multiply \( x \) by \( 10^2 = 100 \). So \( 100x = 82.828282\ldots \)

Step3: Subtract the original equation from the new one

Subtract \( x = 0.828282\ldots \) from \( 100x = 82.828282\ldots \). We get \( 100x - x = 82.8282\ldots - 0.8282\ldots \)
Simplifying the left side: \( 99x \). Simplifying the right side: \( 82 \)

Step4: Solve for \( x \)

We have \( 99x = 82 \), so \( x = \frac{82}{99} \)
We check if \( \frac{82}{99} \) can be simplified. The greatest common divisor of 82 and 99 is 1 (since 82 factors are 2 and 41, 99 factors are 3, 3, and 11; no common factors), so it is in simplest form.

Answer:

\(\frac{82}{99}\)