What is 0.67̅ (0.67 repeating) as a fraction?

To convert the repeating decimal 0.67̅ into a fraction, we can use a straightforward algebraic method.

Let x = 0.67̅. This means that:

x = 0.67676767…

To eliminate the repeating part, we can multiply both sides of the equation by 100 (since the repeating part is two digits long):

100x = 67.676767…

Now we have two equations:

1. x = 0.676767…
2. 100x = 67.676767…

Next, we subtract the first equation from the second:

100x – x = 67.676767… – 0.676767…

This simplifies to:

99x = 67

Now, solving for x gives us:

x = 67 / 99

So, 0.67̅ as a fraction is 67/99.

We can also check if this fraction can be simplified. The greatest common divisor (GCD) of 67 and 99 is 1, which means that 67/99 is already in its simplest form.