The average of 11 numbers is 36. The average of 9 of them is 34. If the remaining two numbers are in the ratio 2 : 3, find the smaller number.

Introduction / Context:
Given averages for the whole and a subset, we can compute the total of the remaining items. With a ratio for those two items, convert the total to parts and extract each number.

Given Data / Assumptions:

• 11-number mean = 36 ⇒ total T = 396.
• 9-number mean = 34 ⇒ subtotal S = 306.
• Remaining two are in ratio 2 : 3.

Concept / Approach:
Sum of the remaining two = T − S. If they are 2k and 3k, then 5k equals that sum; solve k to get both, then choose the smaller (2k).

Step-by-Step Solution:

Verification / Alternative check:
Other number = 54; together 36 + 54 = 90. Totals reassemble correctly: 306 + 90 = 396 ⇒ average 36.

Why Other Options Are Wrong:
18, 54, 48: Do not match the ratio and total simultaneously. 36 is the unique smaller value.

Common Pitfalls:
Dividing 90 by 2 (getting 45, 45) and ignoring the 2 : 3 ratio, or picking the larger number by mistake.

Final Answer:
36