Manhattan GMAT Challenge Problem of the Week – 19 July 2010

by Manhattan Prep, Jul 20, 2010

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As always, the problem and solution below were written by one of our fantastic instructors. Each challenge problem represents a 700+ level question. If you are up for the challenge, however, set your timer for 2 mins and go!

Question

A total of j dollars is given to Abby, Bill, and Carla and divided equally among them. Then Abby gives Bill k dollars, Bill gives Carla 2k dollars, and Carla gives Abby 3k dollars. Afterwards, Abby has exactly half of all the dollars given to the three people. In terms of k, how much money was given originally to Abby, Bill, and Carla?

A. 4k

B. 6k

C. 8k

D. 12k

E. 16k

Answer

Taking a direct algebra approach, we can see that each of the three people receives j/3 dollars. During the swaps, Abby gives away k dollars but receives 3k dollars, so her total increases by 2k dollars. Her final total is half of the original amount of money, or j/2. Now we can write an equation:

j/3 + 2k = j/2

Now solve for j in terms of k. First, multiply through by 6 to eliminate fractions:

2j + 12k = 3j

12k = j

This is our answer. We can also solve by picking a number for j, but realize that we cannot separately pick numbers for j and kafter all, that would determine the very relationship the question is asking us for. Moreover, its difficult to know ahead of time what would be a good test number for j. Seeing 2 and 3 as coefficients within the problem, we might pick $6 as the total. Then each of the people receives $2. Abby has to wind up with $3 (half of $6) when all is said and done, so she has to increase her total by $1. Since she gives away k dollars but receives 3k dollars, she increases her total by 2k dollars. This tells us that k is $0.50, so j is 12 times bigger. However, this reasoning doesnt save us a whole lot of work. Direct algebra is just as fast.

The correct answer is (D).

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