Manhattan Challenge Problem of the week ! - Jan 1

"Smurfs, Elves and Fairies"

One smurf and one elf can build a treehouse together in two hours, but the smurf would need the help of two fairies in order to complete the same job in the same amount of time. If one elf and one fairy worked together, it would take them four hours to build the treehouse. Assuming that work rates for smurfs, elves, and fairies remain constant, how many hours would it take one smurf, one elf, and one fairy, working together, to build the treehouse?

(A) 5/7

(B) 1

(C) 10/7

(D) 12/7

(E) 22/7

Answer : D is the correct answer.

The combined rate of individuals working together is equal to the sum of all the individual working rates.

Let s = rate of a smurf, e = rate of an elf, and f = rate of a fairy. A rate is expressed in terms of treehouses/hour. So for instance, the first equation below says that a smurf and an elf working together can build 1 treehouse per 2 hours, for a rate of 1/2 treehouse per hour.

1) s + e = 1/2
2) s + 2 f = 1/2
3) e + f = 1/4

The three equations can be combined by solving the first one for s in terms of e, and the third equation for f in terms of e, and then by substituting both new equations into the middle equation.

1) s = 1/2 – e
2) (1/2 – e) + 2 (1/4 – e) = 1/2
3) f = 1/4 – e

Now, we simply solve equation 2 for e:

(1/2 – e) + 2 (1/4 – e) = 1/2
2/4 – e + 2/4 – 2 e = 2/4
4/4 – 3e = 2/4
-3e = -2/4
e = 2/12
e = 1/6

Once we know e, we can solve for s and f:

s = 1/2 – e
s = 1/2 – 1/6
s = 3/6 – 1/6
s = 2/6s = 1/3

f = 1/4 – e
f = 1/4 – 1/6
f = 3/12 – 2/12
f = 1/12

We add up their individual rates to get a combined rate:

e + s + f
=1/6 + 1/3 + 1/12
=2/12 + 4/12 + 1/12
= 7/12

Remembering that a rate is expressed in terms of treehouses/hour, this indicates that a smurf, an elf, and a fairy, working together, can produce 7 treehouses per 12 hours. Since we want to know the number of hours per treehouse, we must take the reciprocal of the rate. Therefore we conclude that it takes them 12 hours per 7 treehouses, which is equivalent to 12/7 of an hour per treehouse.

The correct answer is D.