A university library budget committee must reduce exactly five of eight areas of expenditure I, J, K, L, M, N, O and P, under the following conditions: If both I and O are reduced, P is also reduced. If L is reduced, neither N nor O is reduced. If M is reduced, J is not reduced. Of the areas J, K, and N exactly two are reduced. If both K and N are reduced, which pair of areas is such that neither area could possibly be reduced?

Introduction / Context:
This is a classic analytical reasoning or logic game question that involves conditions on a committee reducing certain areas of expenditure. The task is to interpret the rules correctly, apply them systematically, and then answer a specific question about which pair of areas cannot be reduced given an extra condition. Such problems are common in competitive exams to test structured logical thinking.

Given Data / Assumptions:

• There are eight areas of expenditure: I, J, K, L, M, N, O, and P.
• Exactly five of these eight areas are reduced.
• Rule 1: If both I and O are reduced, then P is also reduced.
• Rule 2: If L is reduced, then neither N nor O is reduced.
• Rule 3: If M is reduced, then J is not reduced.
• Rule 4: Of J, K, and N exactly two are reduced.
• Additional condition for the question: Both K and N are reduced.

Concept / Approach:
We must translate the rules into logical constraints and then consider all valid combinations that satisfy them, given that K and N are reduced. From these valid combinations we determine which pair of areas never appears in the list of reduced areas. That pair is the one where neither area could be reduced under the stated conditions.

Step-by-Step Solution:
Step 1: From the condition that both K and N are reduced, and from Rule 4 (exactly two of J, K, N are reduced), we conclude that J cannot be reduced. Thus K and N are reduced, J is not. Step 2: Since N is reduced, Rule 2 tells us that L cannot be reduced, because L being reduced would forbid N from being reduced. Step 3: So far the reduced list must include K and N, and must exclude J and L. Step 4: We still need a total of 5 reduced areas. So three more areas among I, M, O, P can be chosen, subject to the other rules. Step 5: Rule 3 says if M is reduced then J is not reduced. J is already not reduced, so this rule is automatically satisfied and does not add a new restriction here. Step 6: Rule 1 says if I and O are both reduced then P must also be reduced. This simply links I, O, and P but does not affect J or L. Step 7: Now observe that J is never reduced in any valid configuration because K and N are the two chosen among J, K, N. L is also never reduced because N is reduced. Step 8: Therefore the pair of areas that cannot be reduced under these conditions is J and L.

Verification / Alternative check:
We can verify by constructing sample valid combinations. For example: reduce I, K, M, N, P. This satisfies all rules and leaves J and L unreduced. Another valid combination is K, N, O, P, and I, where again J and L are not reduced. In every attempt, N being reduced blocks L, and K and N being the two from J, K, N blocks J. Since we can find valid sets where I, M, O, or P are reduced, but never J or L, the pair J and L is indeed the one that can never be reduced when K and N are reduced.

Why Other Options Are Wrong:

• I and L: I can be reduced in several valid combinations, so this pair is not impossible.
• J and M: J cannot be reduced, but M may be reduced as long as J is not, so the pair is not impossible.
• I and J: I may be reduced; only J is blocked in all valid cases, so the pair is not correct.
• M and O: Both M and O may appear in some valid reduction sets, so this pair is clearly possible.

Common Pitfalls:
Many students forget that 'exactly two' of J, K, N are reduced, which immediately fixes J as not reduced when both K and N are reduced. Others misread Rule 2 and think that N being reduced blocks L from being reduced in the other direction, which is actually correct here. Some learners also forget to count exactly five reduced areas and instead focus only on a subset of rules. A systematic approach, marking each area as forced in or out, helps avoid these mistakes.

Final Answer:
When both K and N are reduced, the two areas that can never be reduced are J and L.