Question

.70 repeating decimal into a simplified fraction

Answer 1

To convert the repeating decimal \(0.70\overline{0}\) (which can be interpreted as \(0.707070...\)) into a simplified fraction, follow these steps:

1. Let \(x = 0.70\overline{0}\).
2. To eliminate the repeating part, multiply \(x\) by 100 (since the repeating part is two digits):
\[
100x = 70.70\overline{0}
\]
3. Now, subtract the original \(x\) from this equation:
\[
100x - x = 70.70\overline{0} - 0.70\overline{0}
\]
\[
99x = 70
\]
4. Now, solve for \(x\):
\[
x = \frac{70}{99}
\]

Now, we'll check if this fraction can be simplified. We can find the greatest common divisor (GCD) of 70 and 99.

The GCD of 70 and 99 is 1 (since 70 is \(2 \times 5 \times 7\) and 99 is \(3^2 \times 11\)), meaning that the fraction is already in its simplest form.

Thus, the repeating decimal \(0.70\overline{0}\) as a simplified fraction is:
\[
\frac{70}{99}
\]