2 men and 5 boys can do a piece of work in 4 days, while 4 men and 4 boys can do it in 3 days.How long it would take one man alone to do it and how many days ity would take one boy to do it alone?

boys rate = 1/b

men's rate = 1/m

job done by 2 men and 5 boys in 4 days
= 4(2/b + 5/m) = 8/b + 20/m

job done by 4 men and 4 boys in 3 days
= 3(4/m + 4/b) = 12/m + 12/b

8/b + 20/m = 12/m + 12/b
8/m = 4/b
8b = 4m
m = 2b or b = m/2
The man's rate is twice the boy's rate.

job = 12/m + 12/b
= 12/(2b) + 12/b = 18/b
= 18(1/b)

One boy alone would take 18 days

Perfect ! Thanks May God bless you :)

To solve this problem, let's first assign variables to represent the work rates of one man and one boy. Let's say the work rate of one man is M units of work per day, and the work rate of one boy is B units of work per day.

Given that 2 men and 5 boys can do the work in 4 days, we can set up the following equation:
2M * 4 + 5B * 4 = 1 (representing the complete work done)

Similarly, for 4 men and 4 boys completing the work in 3 days, we can set up another equation:
4M * 3 + 4B * 3 = 1

Now, we have two equations with two variables. We can solve them simultaneously to find the values of M and B.

Multiplying the first equation by 3 and the second equation by 4 to eliminate the variables M and B, we get:
24M + 60B = 3
12M + 12B = 4

Now, we can solve these two equations using any appropriate method, such as substitution or elimination.

By eliminating M, we multiply the second equation by 2 and subtract it from the first equation:
(24M + 60B) - (24M + 24B) = 3 - 8
36B = 1

Dividing both sides by 36, we find that B = 1/36.
Therefore, the work rate of one boy is 1/36 units of work per day.

Now that we have the value of B, we can substitute it back into one of the original equations to find the value of M.

Substituting B = 1/36 into the first equation, we get:
2M * 4 + 5(1/36) * 4 = 1
8M + 20/36 = 1
8M = 1 - 20/36
8M = 16/36
M = 16/288
M = 1/18

Thus, the work rate of one man is 1/18 units of work per day.

Finally, to find how long it would take one man alone to do the work, we can set up the equation:
(1/18) * D = 1
D = 18 days

So, it would take one man alone 18 days to complete the work.

Similarly, to find how long it would take one boy alone to do the work, we can set up the equation:
(1/36) * D = 1
D = 36 days

Therefore, it would take one boy alone 36 days to complete the work.