A university library budget committee must reduce exactly five of eight areas of expenditure—I, J, K, L, M, N, O and P—in accordance with 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 three areas J, K, and N exactly two are reduced. Question : If both K and N are reduced, which one of the following is a pair of areas neither of which could be reduced?

Given areas: I, J, K, L, M, N, O, P

Total reductions required: Exactly 5 of the 8 areas

Conditions:

1. If I and O are reduced, then P is also reduced.
2. If L is reduced, then neither N nor O is reduced.
3. If M is reduced, then J is not reduced.
4. Exactly two of J, K, N are reduced.

Given in the question: K and N are reduced → So, by the last rule, J is not reduced.

Apply Rule 2: If L is reduced, N or O cannot be reduced. But since N is reduced, L cannot be reduced.

Apply Rule 3: If M is reduced, J is not reduced. But we already know J is not reduced, so M can be reduced.

So far:

• K — reduced
• N — reduced
• L — not reduced
• J — not reduced
• M — can be reduced

We must reduce 5 areas. Already reduced: K, N. Potentially reducible: I, M, O, P.

We want to identify a pair of areas that cannot be reduced. We already know:

• L — cannot be reduced
• J — cannot be reduced

Final Answer: J, L