BTGmoderatorDC wrote: ↑Fri Feb 26, 2021 6:21 pm
If 12 men and 16 women can do a piece of work in 5 days and 13 men and 24 women can do it in 4 days, how long will 7 men and 10 women take to do it?
(A) 4.2 days
(B) 6.8 days
(C) 8.3 days
(D) 9.8 days
(E) 10.2 days
OA
C
Solution:
We can let the time it takes 1 man to finish the work = m, and thus the rate of 1 man = 1/m. Likewise, we can let the time it takes 1 woman to finish the work = w, and thus the rate of 1 woman = 1/w.
Thus, the combined rate of 12 men and 16 women is 12/m + 16/w. Since they can finish the work in 5 days, their combined rate is also equal to 1/5. Thus, we have:
12/m + 16/w = 1/5
Multiplying both sides of the equation by 5mw, we have:
60w + 80m = mw
Similarly, the combined rate of 13 men and 24 women is 13/m + 24/w. Since they can finish the work in 4 days, their combined rate is equal to 1/4. Thus, we have:
13/m + 24/w = 1/4
Multiplying both sides of the equation by 4mw, we have:
52w + 96m = mw
So, we have 60w + 80m = 52w + 96m (since they both equal mw).
60w + 80m = 52w + 96m
8w = 16m
w = 2m
We can now substitute w = 2m into the first equation, 12/m + 16/w = 1/5, to solve for m:
12/m + 16/(2m) = 1/5
12/m + 8/m = 1/5
20/m = 1/5
m = 100
Since m = 100 days, w = 200 days. The rate of 1 man is 1/100 and the rate of 1 woman is 1/200. Thus, the rate of 7 men and 10 women is 7/100 + 10/200 = 7/100 + 5/100 = 12/100, and the time for them to finish the same work is 1/(12/100) = 100/12 = 8.3 days.
Alternate Solution:
Alternatively, we can interpret the equation w = 2m as follows:
The time required for 1 woman to finish the job is twice that of a man, or, in other words, the job done by 2 women is equivalent to the job done by 1 man. We know 12 men and 16 women finish the job in 5 days, and as per the above discussion, this is equivalent to the job done by 12 + 16/2 = 20 men.
The question is asking for the time required to finish the job with 7 men and 10 women working, which is equivalent to 7 + 10/2 = 12 men working. We can set up an inverse proportion to find the required time: If 20 men finish a job in 5 days, then 12 men finish the job in how many days? Letting this unknown quantity be x, we have
20*5 = 12*x
x = 100/12 = 8.3 days
Answer: C
