Certainly! Let's write [tex]\(0.4141\ldots\)[/tex] as a fraction step-by-step.
Let [tex]\( x = 0.4141\ldots \)[/tex].
To express [tex]\( x \)[/tex] as a fraction, we should eliminate the repeating decimal portion. Here is the step-by-step process:
1. Identify the repeating part:
The repeating part of [tex]\( x \)[/tex] is [tex]\( 41 \)[/tex].
2. Multiply [tex]\( x \)[/tex] by a power of 10 that matches the length of the repeating part:
Since the repeating part 41 has 2 digits, we multiply [tex]\( x \)[/tex] by [tex]\( 10^2 = 100 \)[/tex].
[tex]\[
100x = 41.4141\ldots
\][/tex]
3. Set up an equation:
We now have two expressions:
[tex]\[
x = 0.4141\ldots
\][/tex]
[tex]\[
100x = 41.4141\ldots
\][/tex]
4. Subtract the original equation from the multiplied equation:
[tex]\[
100x - x = 41.4141\ldots - 0.4141\ldots
\][/tex]
[tex]\[
99x = 41
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{41}{99}
\][/tex]
Therefore, the fraction representation of the repeating decimal [tex]\( 0.4141\ldots \)[/tex] is:
[tex]\[
\boxed{\frac{41}{99}}
\][/tex]
