Question

Directions: Each question below is followed by two statements, I and II. Decide whether the data in the statements is sufficient to answer the question, using these options:
(AA) Statement I alone is sufficient.
(BB) Statement II alone is sufficient.
(CC) Statements I and II together are sufficient, but neither alone is sufficient.
(DD) Either statement I alone or statement II alone is sufficient.
(EE) Statements I and II together are not sufficient.

A, B, C, D, E and F are six integers such that \(E < F\), \(B > A\), and \(A < D < B\). C is the greatest of the six integers. Is A the smallest integer?
I. \(E + B < A + D\)
II. \(D < F\)

• A. Statement I alone is sufficient.
• B. Statement II alone is sufficient.
• C. Statements I and II together are sufficient, but neither alone is sufficient.
• D. Either statement I alone or statement II alone is sufficient.

Hint:

Rearrange statement I to compare E and A directly, using the already-given fact that D is less than B.

Answer 1

The Correct Option is A

Step 1: Restate the goal as a contradiction check.
Assume, for the sake of argument, that A actually is the smallest of the six integers. Under this assumption, A would have to be less than E as well, in addition to being less than D and B, which are already given. We will check whether each statement is consistent with this assumption.

Step 2: Check statement I under this assumption.
If $A < E$, and we are given $D < B$, adding these two inequalities the same way round gives $A + D < E + B$. But statement I says the opposite, $E + B < A + D$. This is a direct contradiction, so the assumption "A is the smallest" cannot be true whenever statement I holds. That means statement I, by itself, proves A is NOT the smallest, a firm and complete answer. Statement I alone is sufficient.

Step 3: Check statement II with a concrete example.
Statement II only says $D < F$. Try two different sets of numbers that both satisfy every given condition ($E < F$, $B > A$, $A < D < B$, C greatest) plus $D < F$: Example 1: $A=1, D=2, B=5, E=0, F=6, C=10$. Here $A=1 > E=0$, so A is not the smallest. Example 2: $A=1, D=2, B=5, E=3, F=6, C=10$. Here $A=1 < E=3$, so A is the smallest. Both examples satisfy statement II ($D=2<F=6$) and all the given constraints, yet they give opposite answers to "is A the smallest?". So statement II alone cannot pin down a unique answer.

Step 4: Conclude.
Statement I alone always gives a firm "No", while statement II alone can go either way depending on the numbers. So only statement I is independently sufficient, which is option (AA).

Final Answer:
Option (AA), statement I alone is sufficient.