 Work Problems
 Question:
Guna wrote, "I need the method of solving the problems like "a man, a woman and a child together finish a work in 6 days. man alone can finish it within 10 days. woman alone can finish it within 24 days. how many days will the child need to finish the work?"

 Answer:
Guna, this is called a work problem. As written, your problem has no solution, but, the procedure for solving it is the same.
Usually one solves it by writing a specific type of equation.
Before writing the equation a few things must be understood.
 (Work) = (rate)(time),
Work equals the number of hours worked times how much of the job can get done each hour.
If you work for 3 hours and can build 2 toys per hour and , you get (3)(2) = 6 toys built.
 (rate) = (work)/(time)
The rate or speed in which a job is done is the (entire job) divided by (the entire length of time it takes)
or 1/(the time it takes to complete the job).
You can get this equation by taking
(Work) = (rate)(time) and dividing both sides by (time)
(Work)/(time) = (rate)(time)/(time)
(Work)/(time) = (rate)
If you can get 4 toys built in 1 hour, your rate is 4/1 or 4.
If it takes you 4 days to build a big toy, your rate is 1/4 or a quarter of the job done each day.
In your problem the rates are as follow:
man  1/10 because "man alone can finish it within 10 days."
woman  1/24 because "woman alone can finish it within 24 days."
child  1/x because "how many days will the child need to finish the work"
In the second problem the rates are as follow:
tom  1/2 because "tom can paint the room in 2 hours."
dick  1/4 because "dick can paint the room in 4 hours."
 The equation you need is:
(the work that gets done while working together) = (1 whole job)
 You must take this equation and fill in the details with the info from the original problem: "a man, a woman and a child together finish a work in 6 days. man alone can finish it within 10 days. woman alone can finish it within 24 days. how many days will the child need to finish the work?"
(the work that gets done while working together) = (1 whole job)
(work the man does) + (work the woman does) + (work the kid does) = (1 complete job)
(man's rate)(man's time) + (woman's rate)(woman's time) + (kid's rate)(kid's time) = 1
(1/10)(6) + (1/24)(6) + (1/x)(6) = 1
This equation has no solution.
 This new original problem has the following details and equation.
"Tom can paint a room in 2 hours and Dick takes 4 hours to paint the same room. If they work together, how long will it take to paint the room?"
(the work that gets done while working together) = (1 whole job)
(work that Tom does) + (work that Dick does) = (1 complete job)
(Tom's rate)(Tom's time) + (Dick's rate)(Dick's time) = 1
(1/2)(x) + (1/4)(x) = 1
1/(2x) + 1/(4x) = 1 Next, multiply by the common denominator, 4x, to clear the fractions.
2 + 1 = 4x Solve.
3 = 4x
3/4 = x It will take 3/4 of an hour or 45 minutes.
