Zoo Ticket Mix Data Sufficiency
A regional zoo sells adult tickets for $18 and child tickets for $12. Last Saturday the zoo sold only these two ticket types. The operations manager wants to know whether the zoo sold more adult tickets than child tickets, because staffing for the education desk depends on the answer. The manager has two records available: one is the difference in revenue between adult and child tickets; the other is the total number of tickets sold. However, the manager is unsure whether either record alone, or only the two together, can answer the question. The zoo does not record the number of each ticket type separately at the gate, and no complimentary tickets were issued. The manager asks you to evaluate the sufficiency of the records before requesting a manual count.
1. Statement (1): Adult ticket revenue exceeded child ticket revenue by $1,080. Statement (2): A total of 800 tickets were sold. Determine whether the question 'Did the zoo sell more adult tickets than child tickets?' can be answered by statement (1) alone, statement (2) alone, both together, or neither. Show the algebra.
Read the worked answer
Let A = adult tickets and C = child tickets.
Statement (1): 18A - 12C = 1,080, or 3A - 2C = 180. This alone does not fix whether A > C. Example: A = 100, C = 60 gives 300 - 120 = 180 and A > C. Example: A = 200, C = 210 gives 600 - 420 = 180 and A < C. So (1) alone is insufficient.
Statement (2): A + C = 800. This alone gives no comparison between A and C. Insufficient.
Together: substitute C = 800 - A into 3A - 2C = 180: 3A - 2(800 - A) = 180 3A - 1,600 + 2A = 180 5A = 1,780 A = 356, so C = 444.
Thus A < C. The two statements together are sufficient, but neither alone is. Answer: both statements together.
2. Suppose the zoo had recorded instead that adult ticket revenue was exactly twice child ticket revenue. Would that alone be sufficient to answer whether more adult tickets were sold? Explain.
Read the worked answer
Adult revenue = 18A; child revenue = 12C. If adult revenue is exactly twice child revenue, 18A = 2(12C) = 24C, so A = (24/18)C = (4/3)C.
For any nonnegative C, A is either greater than C (when C > 0) or equal to C (when C = 0). In either case the question 'Did the zoo sell more adult tickets than child tickets?' has a definite answer: no if C = 0, yes if C > 0. Since a real ticket day with revenue would have C > 0, A > C. The ratio alone is sufficient; no total ticket count is needed.