BTGmoderatorDC wrote:One woman and one man can build a wall together in two hours, but the woman would need the help of two girls in order to complete the same job in the same amount of time. If one man and one girl worked together, it would take them four hours to build the wall. Assuming that rates for men, women and girls remain constant, how many hours would it take one woman, one man, and one girl, working together, to build the wall?
(A) 5/7
(B) 1
(C) 10/7
(D) 12/7
(E) 22/7
We can let m = the time is takes the man to build the wall, w = the time it takes the woman to build the wall, and g = the time it takes one girl to build the wall. Looking at the rates of these individuals, we see that one man's rate is 1/m, one woman's rate is 1/w, and 1 girl's rate is 1/g. Thus:
1/m + 1/w = 1/2
and
1/w + 2/g = 1/2
and
1/m + 1/g = 1/4
From the first equation, let's isolate 1/m:
1/m = 1/2 - 1/w
Let's substitute this in the equation 1/m + 1/g = 1/4:
1/2 - 1/w + 1/g = 1/4
-1/w + 1/g = -1/4
Adding the equations 1/w + 2/g = 1/2 and -1/w + 1/g = -1/4 together, we obtain:
3/g = 1/4
g = 12
Since it takes a girl 12 hours to finish the job, her rate is 1/12. We are looking for 1/m + 1/w + 1/g; therefore, we add 1/12 to the equation 1/m + 1/w = 1/2:
1/m + 1/w + 1/g = 1/2 + 1/12
1/m + 1/w + 1/g = 7/12
Thus, it will take 1/(7/12) = 12/7 hours for a man, a woman, and a girl to build the wall, working together.
Alternate Solution:
Since the woman can finish the job in the same amount of time with the help of either one man or two girls, the rate of one man is equal to the rate of two girls.
Since one man and one girl can finish the job in 4 hours, and since the rate of one man is equal to the rate of two girls, three girls can finish the job in 4 hours. Since time is inversely proportional to the number of workers, one girl can finish the job in 12 hours.
Since one man and one woman finish the job in two hours, they complete 1/2 of the job in one hour. Since one girl can finish the job in 12 hours, one girl can complete 1/12 of the job in one hour. All working together, they finish 1/2 + 1/12 = 7/12 of the job in one hour. If 7/12 of the job gets done in one hour, then the entire job will get done in 1/(7/12) = 12/7 hours.
Answer: D
