Question

Write the following infinite decimal expression as a fraction you do not need to simplify 0.414141…

Answer 1

To express the repeating decimal \( 0.414141... \) as a fraction, we can use the following method.

Let \( x = 0.414141... \).

Since the repeating part has two digits ("41"), we can multiply \( x \) by \( 100 \) to shift the decimal point two places to the right:

\[
100x = 41.414141...
\]

Now, we subtract the original equation from this new equation:

\[
100x - x = 41.414141... - 0.414141...
\]
\[
99x = 41
\]

Next, we solve for \( x \):

\[
x = \frac{41}{99}
\]

Thus, the repeating decimal \( 0.414141... \) can be expressed as the fraction \( \frac{41}{99} \).