Question

Convert 0.02--------- to a fraction.

Answer 1

To convert the repeating decimal \(0.02\overline{0}\) (where the "0" repeats indefinitely) into a fraction, we can follow these steps:

1. Let \(x = 0.020000...\)
2. To eliminate the repeating part, multiply \(x\) by 100 (because the repeating part is two decimal places):
\[
100x = 2.00000...
\]
3. Now, write down the equation based on our two equations:
\[
100x = 2
\]
\[
x = 0.02
\]
4. Subtract the second equation from the first:
\[
100x - x = 2 - 0.02
\]
\[
99x = 1.98
\]
5. Now, solve for \(x\):
\[
x = \frac{1.98}{99}
\]
6. Next, simplify \(1.98\):
\[
1.98 = \frac{198}{100}
\]
So:
\[
x = \frac{198/100}{99} = \frac{198}{9900}
\]
7. Now, simplify \(\frac{198}{9900}\):
\[
\frac{198}{9900} = \frac{99}{4950}
\]
Both 99 and 4950 can be divided by 9:
\[
\frac{11}{550}
\]
8. Finally, simplifying further gives:
\[
\frac{11}{550} = \frac{1}{50}
\]

Thus, the fraction representation of \(0.02\overline{0}\) is:

\[
\frac{1}{50}
\]