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:
- \( x = 0.404040...\)
- \( 100x = 40.404040...\)
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:
\[ \boxed{\frac{40}{99}} \]
1 answer