*— but it's the technique that they want you to learn, not the applicability to "real life".The method of solution for "work" problems is not obvious, so don't feel bad if you're totally lost at the moment.*

There is a "trick" to doing work problems: you have to think of the problem in terms of how much each person / machine / whatever does Since the unit for completion was "hours", I converted each time to an hourly rate; that is, I restated everything in terms of how much of the entire task could be completed per hour.

To do this, I simply inverted each value for "hours to complete job": My first step is to list the times taken by each pipe to fill the pool, and how long the two pipes take together.

But no one has ever been able to prove that for certain.

It's possible that there's some really big number that goes to infinity instead, or maybe a number that gets stuck in a loop and never reaches 1.

For solutions of mathematical optimization problems, see Feasible solution.

In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equality sign.

In this case, I know the "together" time, but not the individual times.

One of the pipes' times is expressed in terms of the other pipe's time, so I'll pick a variable to stand for one of these times. Since the faster pipe's time to completion is defined in terms of the second pipe's time, I'll pick a variable for the slower pipe's time, and then use this to create an expression for the faster pipe's time: Then I make the necessary assumption that the pipes' contributions are additive (which is reasonable, in this case), add the two pipes' contributions, and set this equal to the combined per-hour rate: Note: I could have picked a variable for the faster pipe, and then defined the time for the slower pipe in terms of this variable.

The thing is, they've never been able to that there isn't a special number out there that never leads to 1.

Mathematicians have tried millions of numbers and they've never found a single one that didn't end up at 1 eventually.

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