Question

Questions 48 to 50 are followed by two statements labelled as (1) and (2). You have to decide if these statements are sufficient to conclusively answer the question. Give answer:

(A) If statement (1) alone or statement (2) alone is sufficient to answer the question
(B) If you can get the answer from (1) and (2) together but neither alone is sufficient
(C) If statement 1 alone is sufficient to answer the question and statement (2) alone is also sufficient
(D) If neither statement (1) nor statement (2) is sufficient to answer the question

A, B, C, D, E are five positive numbers.
\( A + B < C + D \), \( B + C < D + E \), \( C + D < E + A \).
Is 'A' the greatest?

Statement 1: \( D + E < A + B \).
Statement 2: \( E < C \).

• A. If statement (1) alone or statement (2) alone is sufficient to answer the question
• B. If you can get the answer from (1) and (2) together but neither alone is sufficient
• C. If statement 1 alone is sufficient to answer the question and statement (2) alone is also sufficient
• D. If neither statement (1) nor statement (2) is sufficient to answer the question

Hint:

Combine the three given inequalities in pairs to cancel common terms, then check each statement separately against those cancellations; try a concrete counterexample for the statement that seems weaker.

Answer 1

The Correct Option is A

Step 1: Write down what is given.
We are told A, B, C, D, E are all positive, and:
\[ A + B < C + D \quad (i) \]
\[ B + C < D + E \quad (ii) \]
\[ C + D < E + A \quad (iii) \]
We want to know if A is bigger than every one of B, C, D, E.

Step 2: Get one fact for free, before using either statement.
Chain (i) and (iii) together: \( A + B < C + D \) and \( C + D < E + A \), so \( A + B < E + A \).
Cancel A from both sides: \( B < E \). This holds no matter what, straight from the question stem, with no statement needed yet.

Step 3: Test statement 1 alone: \( D + E < A + B \).
Combine this with (i): \( D + E < A + B < C + D \), so \( D + E < C + D \). Cancel D: \( E < C \).
Combine statement 1 with (ii): \( B + C < D + E \) and \( D + E < A + B \) from statement 1, so \( B + C < A + B \). Cancel B: \( C < A \).
We now know \( E < C < A \), so \( E < A \) as well.
From Step 2, \( B < E \), and we just showed \( E < A \), so \( B < E < A \), giving \( B < A \).
For D: statement 1 says \( D + E < A + B \), so \( D < A + B - E \). Since \( B < E \), the quantity \( B - E \) is negative, so \( A + B - E \) is less than A. Combining, \( D < A + B - E < A \), so \( D < A \).
We have now shown all four: \( B < A \), \( C < A \), \( D < A \), \( E < A \). So statement 1 alone is enough to conclude A is the greatest.

Step 4: Test statement 2 alone: \( E < C \).
Try to build a valid, positive-number example using only the original three inequalities plus \( E < C \), and see whether A always comes out biggest.
Take \( A = 10 \), \( C = 12 \), \( D = 5 \), \( E = 8 \), \( B = 0.5 \). Check: \( A + B = 10.5 \), \( C + D = 17 \), so (i) holds. \( B + C = 12.5 \), \( D + E = 13 \), so (ii) holds. \( C + D = 17 \), \( E + A = 18 \), so (iii) holds. Also \( E < C \), since \( 8 < 12 \), so statement 2 holds.
But here \( C = 12 \) is bigger than \( A = 10 \), so A is NOT the greatest in this valid example. This shows statement 2 alone can be true while A is not the biggest number, so statement 2 alone cannot conclusively answer the question.

Step 5: Decide between the answer choices.
Statement 1 alone is enough to prove A is the greatest, in every case satisfying it. Statement 2 alone is not enough, since a valid counter-example exists where A is not the greatest. So exactly one of the two statements, statement 1, works by itself.

Final Answer:
Statement 1 alone answers the question; statement 2 alone does not.
\[ \boxed{\text{Statement (1) alone is sufficient}} \]