A test is scored out of 100 and the scores are divided into five quintile groups. Students are not told their scores, but only their quintile group.

Quantity A	Quantity B
The scores of two students in the bottom quintile group, chosen at random and added together.	The score of a student in the top quintile group, chosen at random.

A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
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Sandy
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Explanation

Since the average is 7, use the average formula to find the sum of the scores in the class:
Average = Sum ÷ (# of terms)
7 = Sum ÷ 20

Sum = 140

At least one student got every possible score. There are eleven possible scores: 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10. This is an evenly spaced set, so calculate the sum by multiplying the average of the set by the number of terms in the set. The average is (10 + 0)/2 = 5 and the number of terms is 11, so the sum of the set is 5 × 11 = 55. Subtract this from the earlier sum; the remaining 9 students had to score 140 – 55 = 85 points.

Quantity A is asks for the lowest score that could have been received by more than one student. If 9 students scored a total of 85 points, and any one student could not score more than 10 points, then what is the lowest possible score that one of these 9 students could have received? In order to
minimize that number, maximize the numbers for the other students. If 8 students scored 10 points each, for a total of 80 points, then the 9th student must have scored exactly 5. Quantity A must be greater than Quantity B. Notice that the average score of 7 forces a lot of the scores to be 10 in order
to to balance out the very low scores of 0, 1, 2, etc., that are required in the class (at least one of each). The lowest score that could have been received by 2 students is 5, so Quantity A is 5.
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Sandy
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Quintiles (“fifths” of the data) define relative scores, not absolute scores. Imagine two possible score distributions:

For instance: The class’s scores are 1, 2, 3, 4, 5 (20% of the class scored each of these). In this case, adding up two lowest quintile students would be 1 + 1 = 2, which is less than 5.

For instance: The class’s scores are 10, 11, 12, 13, 14 (20% of the class scored each of these — still not so sharp!). In this case, adding up two lowest quintile students would be 10 + 10 = 20, which is greater than 14.

Thus, you cannot determine which quantity is larger.

The answer is D.

Regards
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