Question

To convert the repeating decimal 0.82 to a rational number,
you would set x equal to the repeating decimal of 0.82 and then multiply both sides of the equation by what number?

Answer 1

To convert the repeating decimal \(0.82\) (which is interpreted as \(0.828282...\)) to a rational number, you would set \(x\) equal to the repeating decimal:

\[
x = 0.828282...
\]

Since the repeating part is 2 digits long (the "82"), you would multiply both sides of the equation by \(100\) (which is \(10^2\), corresponding to the 2 digits). This gives:

\[
100x = 82.828282...
\]

Next, you can set up a second equation:

\[
x = 0.828282...
\]

Now, you can subtract the first equation from the second:

\[
100x - x = 82.828282... - 0.828282...
\]

This simplifies to:

\[
99x = 82
\]

Now you can solve for \(x\):

\[
x = \frac{82}{99}
\]

So, the number you would multiply both sides by is \(100\).