Question

Convert the following repeating decimal to a fraction in simplest form.
., start overline, 6, 5, end overline
.
65

Answer 1

To convert the repeating decimal \( 0.656565\ldots \) (represented as \( 0.\overline{65} \)) into a fraction, we can set it equal to a variable.

Let:
\[
x = 0.656565\ldots
\]

To eliminate the repeating part, we can multiply both sides of the equation by 100 (since the repeating part has two digits):
\[
100x = 65.656565\ldots
\]

Now, we have two equations:
1. \( x = 0.656565\ldots \)
2. \( 100x = 65.656565\ldots \)

We can now subtract the first equation from the second:
\[
100x - x = 65.656565\ldots - 0.656565\ldots
\]
\[
99x = 65
\]

Now we can solve for \( x \):
\[
x = \frac{65}{99}
\]

Next, we need to ensure that this fraction is in its simplest form. To do this, we can find the greatest common divisor (GCD) of 65 and 99.

The prime factorization of 65 is:
\[
65 = 5 \times 13
\]
The prime factorization of 99 is:
\[
99 = 3^2 \times 11
\]

Since there are no common factors between 65 and 99, the GCD is 1. Therefore, the fraction \( \frac{65}{99} \) is already in its simplest form.

Thus, the repeating decimal \( 0.\overline{65} \) converts to the fraction:
\[
\boxed{\frac{65}{99}}
\]