Problem Solving

swerve wrote:It take 6 days for 3 women and 2 men working together to complete a work. 3 men would do the same work 5 days sooner than 9 women. How many times does the output of a man exceed that of a woman?

A. 3 times
B. 4 times
C. 5 times
D. 6 times
E. 7 times

We can let m = the number of days a man can complete the work by himself and w = the number of days a woman can complete the work by herself. Thus, 1/m = the rate of a man, and 1/w = the rate of a woman.

We can create the equations

6(3 x 1/w + 2 x 1/m) = 1

and

1/(3 x 1/m) = 1/(9 x 1/w) - 5

Simplifying the first equation, we have:

3/w + 2/m = 1/6

Multiplying the equation by 6wm, we have:

18m + 12w = mw

Simplifying the second equation, we have:

m/3 = w/9 - 5

Multiplying the equation by 9, we have:

3m = w - 45

3m + 45 = w

Substituting this in 18m + 12w = mw, we have:

18m + 12(3m + 45) = m(3m + 45)

18m + 36m + 540 = 3m^2 + 45m

3m^2 - 9m - 540 = 0

m^2 - 3m - 180 = 0

(m - 15)(m + 12) = 0

m = 15 or m = -12

Since m can't be negative, m = 15. Hence, w = 3(15) + 45 = 90. We see that a man's rate is 6 times that of a woman's rate since the number of days a man can complete the job is only 1/6 of the number of days a woman can complete the job.

Answer: D