Question

what is 0.07 in the simplest form as a fraction (7 repeats)

Answer 1

To convert the repeating decimal \(0.07\overline{07}\) (where the "07" repeats indefinitely) into a fraction, you can use the following steps:

1. Let \( x = 0.07\overline{07} \).

2. To eliminate the repeating part, multiply \( x \) by \( 1000 \) (since the repeat has 2 digits before it starts repeating):
\[
1000x = 70.7\overline{07}
\]

3. Next, multiply \( x \) by \( 10 \):
\[
10x = 0.7\overline{07}
\]

4. Now, you can set up the equation:
\[
1000x - 10x = 70.7\overline{07} - 0.7\overline{07}
\]
Simplifying this gives:
\[
990x = 70
\]

5. Solving for \( x \):
\[
x = \frac{70}{990}
\]

6. Now, simplify the fraction:
\[
\frac{70}{990} = \frac{7}{99} \quad \text{(dividing both the numerator and the denominator by 10)}
\]

Thus, the repeating decimal \(0.07\overline{07}\) in the simplest form as a fraction is \(\frac{7}{99}\).