Convert the following repeating decimal to a fraction in simplest form.
., start overline, 0, 1, end overline
.
01
To convert the repeating decimal .,\overline{0,1} to a fraction, we can use the fact that .,\overline{0,1} is equal to x, where x = .,\overline{01}.
Let's multiply both sides of the equation x = .,\overline{01} by 1000 to shift the repeating part two places to the left:
1000x = 10,.\overline{01}
Now, subtract the original equation x = .,\overline{01} from 1000x = 10,.\overline{01}:
1000x - x = 10,.\overline{01} - .,\overline{01}
999x = 10
Finally, divide both sides of the equation by 999 to solve for x:
x = 10/999
Therefore, the fraction that represents the repeating decimal .,\overline{0,1} in simplest form is 10/999.