To convert the repeating decimal \( 0.82\overline{82} \) (where "82" is the repeating part) to a rational number, you can set:
\[ x = 0.828282\ldots \]
Next, since the repeating block has two digits ("82"), you multiply both sides of the equation by \( 100 \) (which moves the decimal point two places to the right):
\[ 100x = 82.828282\ldots \]
Now, you can set up an equation to eliminate the repeating decimal:
\[ 100x = 82.828282\ldots \] \[ x = 0.828282\ldots \]
By subtracting the second equation from the first, you have:
\[ 100x - x = 82.828282\ldots - 0.828282\ldots \]
This simplifies to:
\[ 99x = 82 \]
Now, solve for \( x \):
\[ x = \frac{82}{99} \]
So, to summarize, you set \( x = 0.82\overline{82} \) and then multiplied both sides of the equation by \( 100 \).