Question

Convert the following repeating decimal to a fraction in simplest form. 0.40 (40 is repeated)

Answer 1

To convert the repeating decimal \( 0.404040...\) (where "40" is the repeating part) into a fraction, we can use the following method:

Let \( x = 0.404040...\).

We can express this in a way that isolates the repeating part. To do this, we will multiply both sides of the equation by 100 (since the repeating block "40" has 2 digits):

\[
100x = 40.404040...
\]

Now, we have two equations:

1. \( x = 0.404040...\)
2. \( 100x = 40.404040...\)

Next, we can set up a subtraction by subtracting the first equation from the second:

\[
100x - x = 40.404040... - 0.404040...
\]

This simplifies to:

\[
99x = 40
\]

Now we solve for \( x \):

\[
x = \frac{40}{99}
\]

Now, we need to check if the fraction \( \frac{40}{99} \) can be simplified. The greatest common divisor (GCD) of 40 and 99 is 1 (since 40 is \( 2^3 \times 5 \) and 99 is \( 3^2 \times 11 \), they have no common factors).

Thus, the fraction is already in simplest form.

Therefore, the repeating decimal \( 0.404040...\) as a fraction in simplest form is:

\[
\boxed{\frac{40}{99}}
\]