Tasks for collaboration are familiar to school students of many generations. They are quite often offered on a final assessment, however time for their decision in a school course of mathematics is taken away a little. Having understood the principle of the solution of problems of similar types, you will not become puzzled also at examination.

## It is required to you

- - collection of tasks;
- - ability to solve the systems of the equations;
- - knowledge of receptions of the rational account.

## Instruction

1. Define to what subtype the task for collaboration belongs. Main subtypes three. These are tasks on calculation of time, speed of filling of the pool through pipes with a different capacity and also on calculation of the way passed by two or several moving bodies. The last subtype is very similar to tasks on the movement.

2. In a general view a statement of the problem on calculation of time look approximately so. One worker can perform a task quicker, than the second. at a size a. Together they will spend b of hours. It is necessary to find, what is the time it will be required to everyone to execute all amount of works. Take over all work for 1.

3. Time necessary for everyone, designate as x and y. Find productivity of each worker. For this purpose it is necessary 1 to divide into time, that is on x and y.

4. Express the equation how many will make everyone for that time while they work together. For this purpose increase performance 1/x and 1/y for the period of an and put both numbers. Result - all volume of work, that is 1. Thus, the first equation at you will look as and (1/x + 1/y) =1.

5. The second equation of a system will represent a difference between x and y which equals to number b. Solve the system of the equations, having expressed one of unknown through another. For example, y=b-x. Having substituted this value in the first equation of a system, you can calculate x.

6. Conditions of tasks of this kind can differ from each other, but the principle remains to the same. For example, to you it is given that some time two workers worked together, and then one stopped working. Another performed the remained task for some time. Anyway all volume will be equal to 1. In the same way as well as in the first case, designate time of one and the second as x and at. Express performance, having divided work for a while.

7. Express how many each worker made while they worked together, having increased performance by the general time. Then express the volume of work of one executed for the general time through the volume of work of the second and work out the system of the equations.

8. The well-known problems on the pool are solved on the same algorithm, only for 1 it is necessary to accept all volume of water. For the system of the equations it is necessary to express at first how many water joins or pours out from each pipe per unit of time. Then express an amount of water from one pipe through quantity another and solve a system.