Mess Contribution: Two Claims, One Fixed Bill
A hostel warden must fix the monthly mess contribution for the coming term. The hostel's total mess bill is contractually fixed at ₹10,800 per month, and every resident is expected to pay the same full share, with no concessions, arrears, or extra charges. Two committee members examine the same data. Member A claims: 'If 12 more residents join, the full monthly contribution per head will fall by ₹30.' Member B claims: 'The current full monthly contribution is ₹180 per head.' A third member, unconvinced, argues that neither statement is sufficient on its own and that both must be used together to find the number of residents. The warden wants a clear ruling before signing the notice: whether each claim, taken alone, determines the current number of residents; whether the two claims are consistent; and what the per-head contribution would be after 12 more residents join. No other information is available, and the fixed bill is the only cost.
1. Evaluate the third member's claim that both statements are needed together. Which statement or statements are sufficient to determine the current number of residents?
Read the worked answer
Let the current number of residents be n and the current full share be s.
Member B alone: The total bill is fixed at ₹10,800, so n × 180 = 10,800, giving n = 60. Statement B alone is sufficient.
Member A alone: The current share is 10,800/n. After 12 more residents join, the share is 10,800/(n+12), which is ₹30 less:
10,800/n − 10,800/(n+12) = 30
10,800[(n+12) − n] / [n(n+12)] = 30
129,600 = 30n(n+12)
n(n+12) = 4,320
n² + 12n − 4,320 = 0
(n − 60)(n + 72) = 0
Since n must be positive, n = 60. Statement A alone is also sufficient.
Conclusion: The third member is wrong. Each statement alone determines n = 60; using both together is unnecessary. In data-sufficiency terms, the answer is 'either statement alone is sufficient', not 'both together are necessary'.
2. Are the two claims consistent with each other? Show the check explicitly.
Read the worked answer
Yes. Member B says the current share is ₹180. With n = 60 residents and a fixed bill of ₹10,800, the share is 10,800 ÷ 60 = ₹180. Member A's condition also gives n = 60. Both claims therefore describe the same hostel: 60 residents paying ₹180 each.
A consistency check matters because two statements can each be sufficient yet give different values. Here they agree. If Member B had claimed a share of ₹200, for example, B alone would give n = 54, while A alone would still give n = 60; the statements would then be individually sufficient but mutually contradictory. The warden should not accept a notice based on contradictory data without checking the bill and resident count.
3. Using the validated data, what is the per-head contribution after 12 more residents join? Does this answer depend on which statement is used?
Read the worked answer
After 12 more residents join, the number becomes 60 + 12 = 72. The fixed bill remains ₹10,800, so the new full share is:
10,800 ÷ 72 = ₹150 per head.
The answer does not depend on which statement is used, because both statements were shown to yield the same current resident count. Member A's claim can also be checked directly: the share falls from ₹180 to ₹150, a fall of ₹30, which matches the claim. The warden can sign the notice with the new figure of ₹150, provided the fixed bill and the arrival of exactly 12 residents are both confirmed.