Convert the recurring decimal 0.1\u0305\u030536 (with 36 repeating after the initial 1) to a fraction in lowest terms. Identify the exact rational form.

Introduction / Context:
Recurring decimals that include a non-repeating part followed by a repeating block can be converted to fractions using a standard algebraic technique. Here, the decimal is 0.1363636… where “36” repeats after an initial non-repeating “1”.

Given Data / Assumptions:

• Let x = 0.1363636… with non-repeating part length m = 1 (the digit “1”) and repeating block length n = 2 (“36”).

Concept / Approach:
Use the formula derived from shifting decimals: If x = N.R. + R (repeating), then 10^(m+n) x − 10^m x eliminates the repeating tail. The fraction is then an integer over 9…90…0 with n nines and m zeros.

Step-by-Step Solution:
Let x = 0.1363636…100x = 13.636363… (shift by two for “36”).10x = 1.363636… (shift by one for the non-repeating “1”).Subtract: 100x − 10x = 13.636363… − 1.363636… = 12.273 = 12.273?? (better use integer blocks).Cleaner approach: write digits: up to first repeat: 0.1 | 36 36 36… ⇒ numerical block method gives (136 − 1) / (99 × 10) = 135 / 990.Reduce: 135/990 = 27/198 = 3/22.

Verification / Alternative check:
3/22 ≈ 0.13636… Exactly matches the repeating pattern with “36”.

Why Other Options Are Wrong:

• 136/999 and 136/990 mis-handle the lengths of the non-repeating and repeating parts.
• 136/1000 corresponds to the terminating decimal 0.136, not repeating.
• 13/90 is a simplified-looking fraction but does not reproduce 0.13636…

Common Pitfalls:
Forgetting to include a zero after the n nines to account for the m non-repeating digits or failing to reduce the final fraction completely.

Final Answer:
3/22