Question

Convert the following repeating decimal to a fraction in simplest form.
., start overline, 6, 7, end overline
.
67

Answer 1

To convert the repeating decimal \( 0.\overline{67} \) to a fraction, we can use the following steps:

Let \( x = 0.\overline{67} \).

This means that:

\[
x = 0.67676767\ldots
\]

Next, we multiply both sides of the equation by 100 to move the decimal point two places to the right:

\[
100x = 67.67676767\ldots
\]

Now we have the two equations:

1. \( x = 0.67676767\ldots \)
2. \( 100x = 67.67676767\ldots \)

We can subtract the first equation from the second:

\[
100x - x = 67.67676767\ldots - 0.67676767\ldots
\]

This simplifies to:

\[
99x = 67
\]

Now, divide both sides by 99:

\[
x = \frac{67}{99}
\]

To ensure that the fraction is in simplest form, we check if 67 and 99 have any common factors.

- The number 67 is a prime number.
- The factors of 99 are \( 1, 3, 9, 11, 33, 99 \).

Since 67 is not a factor of 99, \( \frac{67}{99} \) is already in simplest form.

Thus, the repeating decimal \( 0.\overline{67} \) as a fraction in simplest form is:

\[
\boxed{\frac{67}{99}}
\]