Difficult Work/Rate Problem

Tough and ugly problem, I'd love to know a faster way.

But algebraically here's how to do it:

Rate + Rate = Total Constant Rate

Remember that rate=job/day (time is always on the bottom)

m=rate of a man
b=rate of a boy
w=rate of a woman

So we know that

13(m)+7(b)=1/7
and
6(b) + 13(w)=1/6

Solve for m and w respectively so that we can calculate everything in terms of b.

m=(1/7-7b)/13
w=(1/6-6b)/13
b=b

We want to know what happens when we have one man, one woman and one boy

So

The combined constant rate = (1/7-7b)/13 + (1/6-6b)/13 + b
Simplify (all the b's cancel out) and get

(1/7 + 1/6) / 13

Find common denominators

This is where you want to keep everything in fractions til the end.

(1/7 + 1/6) / 13 = (1/13)*(1/7 + 1/6) =(1/13)*(6/(42) + 7/42)= (1/13)*(13/42) = 1/42 is the combined rate of one man, one boy, and one woman which verbally translates to

1 job per 42 days.

Answer A

13 Men and 7 Boys finish work in 7 days.

20 people finish work in 7 days.

20 people work for total of 140 days.

1 people work for 7 days.

13 Women and 6 Boys finish work in 6 days.

19 people finish work in 6 days.

19 people work for total of 114 days.

1 people work for 6 days.

Since the rate is same for all 3, 1 person(men or women or boy) fiinsh the job in 42 days.

13M+7B=1 job in 7 days.
6B+13W=1 job in 6 days.

13M+7B=(1/7)th job in 1 day
6B+13W=(1/6)th job in 1 day
________________________
13M+13B+13W=(1/7+1/6)th job in 1 day

-> 13(M+B+W)=(13/42)th job in 1 day
-> M+B+W=(1/42)th job in 1 day

Thus to complete (1/42)th job = 1 day
and to complete 1 job = ? days
-> 42 days is the answer

Hope this work helps!!!

mattocks wrote:13 men and 7 boys can finish a job in 7 days, while 6 boys and 13 women can finish the same job in 6 days. In how many days can 1 man, 1 boy, and 1 woman working together finish the same job (assume constant rates for each men, women, and boys)?

A) 42
B) 54
C) 21
D) 63
E) 24

When the job is undefined, we can plug in our own values for the job.

13 men and 7 boys can finish a job in 7 days:

m = 1 unit/day and b = 1 unit/day.
13m + 7b = 13(1) + 7(1) = 20 units/day.
Over 7 days, 7 * 20 = 140 total units.

6 boys and 13 women can finish the same job in 6 days:

6b = 6 units/day.
Over 6 days, 6b = 6 * 6 = 36 units.
13w = 140 - 36 = 104 units.
13w = 104/6 = 52/3 units/day.
w = (52/3)/13 = 4/3 units/day.

So m + b + w = 1 + 1 + 4/3 = 10/3 units/day.

So 140 units would take 140/(10/3) = 42 days.

The correct answer is A.