josemarioamaya wrote:

Two teams are distributing information booklets. Team A distributes 60% more boxes of booklets than Team B, but each box of Team A’s has 60% fewer booklets than each box of Team B’s. Which of the following could be the total number of booklets distributed by the two groups?

A. 2,000

B. 3,200

C. 4,100

D. 4,800

E. 4,900

Official Solution (Credit: Manhattan Prep)

This problem is disguising several things, but a few key clues will help to decipher the best solution. First, the question asks which of the answers could be the total. More than one number, then, will satisfy the described conditions, but all of the numbers that do fulfill the conditions will need to share some characteristic.

What is that characteristic? Try to find the simplest number that will actually satisfy the details of the problem; use that to help decipher the needed characteristic. Team A distributes 60% more boxes than Team B. If Team B distributes 10 boxes, then Team A distributes 16. Simplify those numbers even more: if team B distributes 5 boxes, then Team A distributes 8.

Are you getting any ideas about what the problem is really testing? Ratios! The question looks like a percentage problem to start, but it’s actually a ratio problem. If you pick this up from the start, then you'll know the ratio for 60% is 5 : 8. The ratio of Team A boxes to Team B boxes is 8 : 5. (Remember that A has more boxes.)

Next, one box from Team A contains 60% fewer booklets than one box from Team B. That's the same as saying that Team A's boxes contain 40% of the number contained in Team B's boxes. 40% is equivalent to a ratio of 2/5, or 2 : 5. For booklets per box, the ratio of Team to Team B is 2 : 5.

At the lowest level in the ratio, then, Team A distributes 8 boxes containing 2 booklets each, for a total of 16 booklets distributed. Team B distributes 5 boxes containing 5 booklets each, for a total of 25 booklets distributed. Together, the two teams distribute 16 + 25 = 41 booklets. This is the smallest possible value for the number of booklets distributed.

What would happen if we increased the numbers? Because everything has to happen according to the given ratios, and because the number of boxes and the number of booklets must be integers, the total number of booklets must be a multiple of 41.

Not sure why this always works? Combine the ratios. To get the total number of booklets, multiply the number of boxes by the number of booklets for each Team and then add the totals.

Team A boxes : Team B boxes = 8 : 5

Team A booklets per box : Team B booklets per box = 2 : 5

Total Team A booklets : Total Team B booklets = (8)(2) : (5)(5) = 16 : 25

The sum of that last ratio is 16 + 25 = 41.

Test the answers. Only 4,100 is a multiple of 41.

The correct answer is C.

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