Question

8 men and 12 boys can finish a piece of work in 5 days while 6 men and 8 boys can finish it in 7 days.find the time taken by 1 man alone and that by 1 boy alone to finish the work.

Answer 1

Solutions 

Let's call M the number of days it takes ONE man to do the job alone. 
And B the number of days it takes ONE boy to do the job alone. 
This means the man's rate is 1/M of the job per day per man. 
The boy's rate is similarly 1/B of the job per day per boy. 
You may recall the commonly used equation D = R*T. We also have WORK = R*T.

Given:
8 men and 12 boys finish in 5 days
6 men and 8 boys finish in 7 days

For these two jobs, we can write
m*8*5 + b*12*5 = 1[job]
m*6*7 + b*8*7 = 1[job]
80m + 120b = 1

84m + 112b = 1
m + 3b/2 = 1/80 -> m = 1/80 - 3b/2
3m + 4b = 1/28 -> 3(1/80 - 3b/2) + 4b = 1/28

Solve for b:
3/80 - 9b/2 + 4b = 1/28
b/2 = 3/80 - 1/28 = 21/560 - 20/560 = 1/560
b = 1/280 [jobs/boy/day]

If one boy is working alone, the rate is 1 job per 280 days, so it would take 280 days to finish

The men's rate = m = 1/80 - (3/2)(1/280) = 7/560 - 3/560 = 4/560 = 1/140
Thus a man working alone would need 140 days to complete the job

Answer 2

One boy = 280 days
One men = 140 days.