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### A Simple Formula?

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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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Associated Topics:
High School Basic Algebra
Middle School Algebra
Middle School Word Problems

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