Question

what's .67 (all of its repeating) as a fraction in simplest form?

Answer 1

To convert the repeating decimal 0.67... (where the "67" repeats indefinitely) into a fraction, we can use the following method:

Let \( x = 0.676767...\)

To eliminate the repeating part, we can multiply \( x \) by 100 (since the repeating part has 2 digits):

\[
100x = 67.676767...
\]

Now, we can subtract the original equation from this new equation:

\[
100x - x = 67.676767... - 0.676767...
\]

This simplifies to:

\[
99x = 67
\]

Now, solve for \( x \):

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

Next, we check if the fraction \(\frac{67}{99}\) can be simplified. The greatest common divisor (GCD) of 67 and 99 is 1, which means this fraction is already in its simplest form.

So, the repeating decimal \( 0.67... \) as a fraction in simplest form is:

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