Question

To convert 0.82 (82 is repeated) to a rational number, you would set x equal to 0.82 and then multiply both sides of the equation by what number?

Answer 1

To convert the repeating decimal \( x = 0.828282...\) (where 82 is repeated) to a rational number, you would multiply both sides of the equation by 100. This is because the repeating block of digits "82" has two digits.

Here’s how it works step-by-step:

1. Set \( x = 0.828282...\)
2. Multiply both sides by 100:
\[
100x = 82.828282...
\]
3. Now, notice that the right side \( 82.828282...\) can be rewritten as:
\[
100x = 82 + x
\]
4. Rearranging gives:
\[
100x - x = 82
\]
\[
99x = 82
\]
5. Finally, divide both sides by 99:
\[
x = \frac{82}{99}
\]

So, to sum up, you multiply both sides of the equation by **100**.