Question

To convert 0.264 repeated to a rational number you would multiply both sides by what?

Answer 1

To convert the repeating decimal \( 0.264\overline{264} \) (where "264" repeats indefinitely) to a rational number, you typically follow these steps:

1. Let \( x = 0.264264264\ldots \).

2. Since the repeating part has 3 digits ("264"), multiply both sides by \( 1000 \) (which shifts the decimal point three places to the right):
\[
1000x = 264.264264264\ldots
\]

3. Now, you can set up the equation:
\[
1000x = 264 + x
\]

4. Rearrange to isolate \( x \):
\[
1000x - x = 264 \implies 999x = 264
\]
\[
x = \frac{264}{999}
\]

Thus, to convert \( 0.264\overline{264} \) to a rational number, you multiply both sides by \( 1000 \).