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: Note what is already fixed.
We are already told \(A < D < B\), so D is strictly less than B, meaning \(B - D > 0\). We are also told C is the greatest integer, and \(E < F\). The real question is whether A is smaller than every other integer, in particular whether A is smaller than E, since A is already known to be smaller than D and B.

Step 2: Test statement I alone.
Statement I says \(E + B < A + D\). Rearranging by moving B and D across:
\[ E - A < D - B \]
Since \(D < B\) (already given), the right side \(D - B\) is negative. So:
\[ E - A < D - B < 0 \implies E < A \]
This tells us directly that E is smaller than A, so A cannot be the smallest integer. That is a definite, complete answer, so statement I alone is sufficient.

Step 3: Test statement II alone.
Statement II says \(D < F\). But we are already given \(A < D\), so this only tells us \(A < D < F\), which adds nothing about how A compares to E. Since E is the only integer not yet placed relative to A, and statement II is silent about E entirely, we cannot say whether A is the smallest or not. Statement II alone is not sufficient.

Step 4: Conclude.
Only statement I alone answers the question, with a definite "No", while statement II alone leaves it open. Since exactly one statement is independently sufficient, and it is statement I, the answer is option (AA), not option (DD), which would need both statements to work alone, and not option (CC), because statement I on its own already gives a complete answer without needing statement II.

Final Answer:
Statement I alone is sufficient; option (AA).