what's .46 (it's all repeating) as a fraction in simplest form

Answer 1

To convert the repeating decimal \(0.46\overline{46}\) (which means \(0.464646...\)) into a fraction, we can use algebra.

Let's denote \(x = 0.464646...\).

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

\[ 100x = 46.464646... \]

Now we have two equations:

1. \(x = 0.464646...\)
2. \(100x = 46.464646...\)

Next, we can subtract the first equation from the second:

\[ 100x - x = 46.464646... - 0.464646... \]

This simplifies to:

\[ 99x = 46 \]

Now, solving for \(x\):

\[ x = \frac{46}{99} \]

Finally, we need to check if this fraction can be simplified. The greatest common divisor (GCD) of 46 and 99 is 1, so the fraction is already in simplest form.

Thus, \(0.46\overline{46}\) as a fraction in simplest form is:

\[ \frac{46}{99} \]