Convert the following repeating decimal to a fraction in simplest form.

., start overline, 0, 7, end overline
.
07

1 answer

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:

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