Flow:

A frequently featured type of a problem has to do with the time it takes to perform a certain task or the time it takes for a certain event to happen. This may take the form of water escaping from a tank or a pool by one exhaust pipe or two or three and all of these may have a different rate of flow. Similarly, the problem may involve the time it would take one person to perform a certain task, while another person may complete the same job in less time (or more). The question posed is generally how long will it take both people to finish the job and how long it will take for the pool to be emptied, if all three hoses will be used.

As an example, let us consider a pool that may be drained by either one of two hoses or by both hoses together. The narrow hose will empty the pool in 24 hours and the wide hose will empty the pool in 12 hours. How long will it take to empty the pool, if both hoses are used together?

In order to calculate the amount of time necessary to empty the pool with both hoses, it is obviously necessary to add up the flow of water through one hose with the flow of water through the other hose, but how can we do that, give the information provided? Obviously, we need a common unit of measure for both hoses, otherwise how could we add them up?

What common unit of measure could we find? Well, if we could add the amount of water that escapes by one hose to the amount of water that escapes through the other, that would give us a total of both hoses. But can we do that?

Yes we can. If you were told that it will take 2 hours to empty the pool, how much of the pool do you think is emptied in 1 hour? One half. We divide 1 (the full pool) by 2 (the number of hours it will take to empty the pool) and the result - ½ - states that each hour, ½ of the pool is emptied. If it would take 3 hours, we would divide 1 by 3 to calculate that 1/3 of the pool is emptied each hour. If it would be 4 hours, we would divide 1 by 4, to figure out that each hour 1/4 of the pool is emptied. Since we know that in our example, one hose will take 12 hours to empty the pool, while the other one would take 24 hours, we can figure out that in one hour, the first hour will empty 1/12 and the other hose 1/24 of the pool

Now if we add up the two fractions, we will get:

Together, the two hoses will each hour empty 1/8 of the pool and therefore will take 8 hours to empty the pool completely.

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