 # Combined Rates

If each of a man's three sons works 1/8 as fast as he does, and the man does a job in 3 hours, how many hours does it take his sons working together to do the job?

(A) 5
(B) 6
(C) 7
(D) 8
(E) 9

Choice D
Level of Difficulty 1 2 3 4 5

This problem is best set up as a combined rates problem.

If the man does the whole job in 3 hours, we can express it as 3 hours/job. Let's invert that fraction (along with its units) and get that the man works at a rate of 1/3 jobs/hour. Since we are told that the sons work at a rate 1/8 as fast as the father, each son works at a rate of (1/8)(1/3) = 1/24 jobs/hour.

Let t be the number of hours it will take all three sons to do the job together. Then (1/24 jobs/hour)(t hours) should give the fractional part of the job that each son does in time t when they work together. The sum of those fractions for all three sons should lead to a value of 1, representing the whole job:

(1/24 jobs/hour)(t) + (1/24 jobs/hour)(t) + (1/24 jobs/hour)(t) = 1

If you solve this equation for t, you will find that t = 8 hours, or choice D.

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