Multiple choice

Directions: The following problem contains a question and two statements giving certain data. You have to decide whether the data given in the statements are sufficient for answering the question. During a 6-day local trade show, the least number of people registered in a single day was 80. Was the average (arithmetic mean) number of people registered per day for the 6 days greater than 90? (1) For the 4 days with the greatest number of people registered, the average (arithmetic mean) number registered per day was 100. (2) For the 3 days with the smallest number of people registered, the average (arithmetic mean) number registered per day was 85.

  1. Statement (1) alone is sufficient, but statement (2) alone is not.

  2. Statement (2) alone is sufficient, but statement (1) alone is not.

  3. Both (1) and (2) together are sufficient, but none of them alone is sufficient.

  4. Both of them independently are sufficient.

  5. Both the statements (1) and (2) together are not sufficient.
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A Correct answer
Explanation

Let the 6 days be d1 <= d2 <= d3 <= d4 <= d5 <= d6. We know d1 >= 80. Statement 1: (d3+d4+d5+d6)/4 = 100, so d3+d4+d5+d6 = 400. Sum of all 6 days = d1+d2+d3+d4+d5+d6 = d1+d2+400. Since d1, d2 >= 80, sum >= 160+400 = 560. Average = 560/6 = 93.33 > 90. Sufficient.

AI explanation

From statement 1, the total number of people registered for the 4 days with the greatest attendance is 4 times 100, which is 400. Because the least number of people registered on any single day is 80, the maximum possible number of people for the remaining 2 days is 80 plus 80, totaling 160. The maximum total attendance for all 6 days would be 400 plus 160 which equals 560, and 560 divided by 6 is approximately 93.3, which is greater than 90. Statement 1 proves the average is always greater than 90, but statement 2 does not account for the 3 days with the highest attendance, so it is insufficient. Statement 1 alone is sufficient, but statement 2 alone is not.

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