Problem Solving

NandishSS wrote:Each of the 15 boxes in a certain warehouse is either 5 pounds or 10 pounds, and the average (arithmetic mean) weight of the boxes in the warehouse is 8 pounds. If the average weight of the boxes in the warehouse is 8 pounds also by removing 5 boxes, how many 10-pound boxes must be removed?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

Each of the 15 boxes in a certain warehouse is either 5 pounds or 10 pounds, and the average (arithmetic mean) weight of the boxes in the warehouse is 8 pounds.
15 boxes have an average weight of 8 pounds.
So, the TOTAL weight = (15)(8) = 120 pounds.

IMPORTANT: If there were an EQUAL number of 5-pound boxes and 10-pound boxes, the average weight WOULD BE 7.5 pounds (i.e., the average of 5 pounds and 10 pounds). HOWEVER, the average weight is 8 pounds, which is closer to 10 pounds than it is to 5 pounds. This tells us there are more 10-pound boxes than 5-pound boxes.

So, let's TEST some possible scenarios:

8 10-pound boxes and 7 5-pound boxes
Does this yield a TOTAL weight of 120 pounds?
(8)(10) + (7)(5) = 80 + 35 = 115 . . . CLOSE BUT NOT QUITE

9 10-pound boxes and 6 5-pound boxes
Does this yield a TOTAL weight of 120 pounds?
(9)(10) + (6)(5) = 90 + 20 = 120 PERFECT!!

So, we start with 9 10-pound boxes and 6 5-pound boxes
IMPORTANT ASIDE: the ratio of 10-pound boxes to 5-pound boxes = 9 : 6 = 3 : 2

The average weight of the boxes in the warehouse is 8 pounds also by removing 5 boxes, how many 10-pound boxes must be removed?
NOTICE that the average weight of the boxes is 8 pounds BEFORE removing the five boxes, and the average weight is ALSO 8 pounds AFTER removing the 5 boxes. So, it must be the case that the five boxes we removed must also have an average weight of 8 pounds.

Well, we already learned that when the ratio of 10-pound boxes to 5-pound boxes is 3 : 2, the average weight is 8 pounds.

So, among the five boxes we remove, 3 must be 10-pound boxes, and 2 must be 5-pound boxes

Answer: C

Hi NandishSS,

Since the average weight of the boxes will stay the SAME (8 pounds) even after 5 boxes are removed, the average weight of the 5 boxes must ALSO be 8 pounds.

With an average weight of 8 pounds, the 5 pounds must have a TOTAL weight of (5)(8) = 40 pounds. Since we're dealing with just 5-pound and 10-pound boxes, you might find it easiest to just 'brute force' the possibilities....

5 10-pounders = 50 pounds
4 10-pounds and 1 5-pounder = 45 pounds
3 10-pounds and 2 5-pounders = 40 pounds

The prompt asks for the number of 10-pounders in the group of 5.

Final Answer: C

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