2012-02-27 09:00:40

If q < r < s < t, and the average (arithmetic mean) of q, r, s, and t is 6, which of the following MUST be true?

I. q + t = r + s
II. q < 5 and t > 8
III. q + r + s + t = 24

A) I only
B) III only
C) I and III
D) II and III
E) I, II, and III

The correct answer is B

Explanation: It's a "must be" prompt, so if you can find one instance that disproves the rule, you can eliminate all answers choices that involve that statement. Take them one at a time.

I. q + t = r + s. This could be true, but must it be? Let's pick some strange numbers and see what we come up with. We know from the prompt that q > r > s > t and that the four numbers must add up to 24. It doesn't say anything about the numbers being positive though. Let's pick -4 for q and 0 for r, both of which work, so long as t + s = 28, then. So let s = 1 and t = 27. Plug these in: q + t = r + s becomes -4 + 27 = 0 + 1. -23 does not equal 1, so Roman Numeral I does not hold true for all numbers.

II. q < 5 and t > 8. So far, with the number we've picked, this holds true. But, let's try something different. Let's say q = 5. If we pick 5 and it works, we can disprove II since it requires q to be less than 5. Let's also pick 7 for t. 5 + 7 = 12. Could r + s = 12? Yes, but only if we pick non-integer numbers for them. Let's say r = 5.5 and s = 6.5. Everything works,but we've found numbers to disprove Roman Numeral II.

III. q+ r + s + t = 24.  Since the average of q, r, s, and t is 6, that means that the sum of q, r, s, and t has to be 24 as average = sum of  parts/ # of parts, and we have 4 parts, so 4 x 6 = 24. III says the sum of the four parts is 24, and this has to be true.

Since only Roman Numeral 3 MUST be true, choice B is correct.