Question

Convert the following repeating decimal to a fraction in simplest form.
., start overline, 0, 7, end overline
.
07

Answer 1

Let \( x = 0.\overline{07} \), which means that \( x = 0.070707... \).

To convert this repeating decimal into a fraction, we can eliminate the repeating part by multiplying \( x \) by a power of 10 that will shift the decimal point to the right, just past the repeating section.

Since "07" has two digits, we multiply \( x \) by \( 100 \):

\[
100x = 7.070707...
\]

Now we have two equations:

1. \( x = 0.070707... \)
2. \( 100x = 7.070707... \)

Next, we subtract the first equation from the second:

\[
100x - x = 7.070707... - 0.070707...
\]

This simplifies to:

\[
99x = 7
\]

Now, we solve for \( x \):

\[
x = \frac{7}{99}
\]

Next, we need to simplify \( \frac{7}{99} \). Since \( 7 \) is a prime number, we check if it divides \( 99 \):

The prime factorization of \( 99 \) is \( 3^2 \times 11 \). Since \( 7 \) does not divide \( 99 \), \( \frac{7}{99} \) is already in its simplest form.

Thus, the repeating decimal \( 0.\overline{07} \) can be expressed as the fraction:

\[
\boxed{\frac{7}{99}}
\]