A, B and C entered into a partnership. After eight months, B and C left the…

Concept

This is a data-sufficiency problem. A statement (or a set of statements) is SUFFICIENT to find the TOTAL annual profit only if it pins the profit down to a single, valid rupee amount. In a partnership each partner's profit share is proportional to (capital x time invested). Here A stays for the full 12 months while B and C stay only 8 months, so the share ratio is 12a : 8b : 8c. A ratio alone never gives a rupee total — you also need at least one absolute money figure, and every share must come out positive for the partnership to be real.

Translating each statement

• A gives absolute differences: profit(B) = profit(A) + 400 and profit(B) = profit(C) + 200. So profit(A) = profit(B) − 400 and profit(C) = profit(B) − 200 — everything in terms of one unknown, profit(B).

• B fixes the capital relation c = (a + b)/2, which only shapes the 12a : 8b : 8c ratio — no rupee amount.

• C gives a pure ratio: profit(C) : [profit(A) + profit(B)] = 3 : 8 — again no rupee amount.

Testing every combination

1. B + C: both are ratio-only. Combined they give the share ratio 6 : 2 : 3, but with no rupee anchor the total stays a multiple of an unknown — the amount cannot be found.

2. A + C: solving 8·profit(C) = 3·[profit(A) + profit(B)] with A's differences gives profit(B) = 200, hence profit(A) = −200 and profit(C) = 0. A negative and a zero share are impossible for a real partnership — no valid total.

3. A + B: converting B's capital relation into shares and applying A's two differences forces a = 0 (one partner invests nothing), which is impossible — no valid total.

4. All three together inherit the same A + B contradiction, so they are inconsistent as well.

Conclusion

No single statement and no combination of them yields a unique, valid, positive total profit. Therefore the data given is insufficient — the total annual profit cannot be determined from these statements.

Cross-check

Notice each pair fails for a different structural reason: the ratio-only pair (B + C) lacks a money figure, while the pairs that include A (A + C, A + B) force an impossible share (negative, zero, or a zero-investment partner). Independent failure modes confirm the data is genuinely insufficient.