A Simple Formula?Date: 02/21/97 at 19:19:53 From: Ross Mannell Subject: Why a simple formula works When looking at a maths problem of the form, "If one person takes two hours to paint a wall and another only one hour, how long will it take the two to paint the wall together?" I have been told that this simple formula gives the result: a + b ----- a * b I have wondered why this is so. Would the following apply for three workers? a + b + c --------- a * b * c Forgive my ignorance, but I am plagued by the need to know. Ross Mannell Date: 02/22/97 at 03:09:39 From: Doctor Mike Subject: Re: Why a simple formula works Hi Ross, If the a and b of your formula mean how many hours it takes each person, like a = 2 and b = 1, then the formula gives 3/2 or 1 hour and 30 minutes. This is not reasonable, since one of the painters alone could do it in 1 hour. Actually, your formula is UPSIDE DOWN. It should be (a*b)/(a+b). Here's why. Let h be the number of hours it takes for the two of them to do it together. The first person could do it all in a hours, but can paint only a fraction of the wall working h hours. Assuming a constant rate of painting, that fraction is h/a. That is, the first painter will paint h/a of the wall in h hours. Similarly, the second painter will paint h/b of the wall in that time. So, when will they be done? When the whole wall is painted, which happens whenever those 2 fractions add up to 1. You need to solve the following equation: (h/a) + (h/b) = 1 h*b + h*a = a*b Multiplying through by a*b h*(b+a) = a*b Factoring out h h = (a*b)/(a+b) Dividing both sides by b+a Okay, now say 3 painters could do it in a hours, b hours or c hours, respectively. Let h be the number of hours they all must work to get it all painted. The first painter finishes h/a of the wall in that amount of time. The other 2 painters manage to paint h/b of the wall and h/c of the wall in that time. Since h is the time to finish the job, all 3 of these fractions must add up to the whole wall, or 1. So you have the equation: h h h --- + --- + --- = 1 a b c Solve this for h (start by multiplying through by a*b*c) and you will get the generalized formula for 3 painters. It's not the formula you wrote above but I think you can finish it off. I hope this helps. -Doctor Mike, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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