Combining Rates of Work

Date: 08/30/98 at 17:05:34
From: REGINA
Subject: Combining Rates of Work

It takes me 3 hours to paint a house. It takes you 5 hours to paint a 
house. How long will it take for both of us to paint a house? I first 
added. Then I changed and multiplied instead. What should I do?

Date: 08/31/98 at 09:32:51
From: Doctor Stacey
Subject: Re: Combining Rates of Work

Hi Regina,
     
Thanks for asking Dr. Math! This is a tricky question. I'm going to 
give you some hints, and then maybe you can do it by yourself.  

First, if it takes you three hours to paint a house, what fraction of 
the house can you paint in one hour?  This is your rate of painting.  
Do the same thing for me. If it takes me five hours to finish the 
house, what fraction of the house can I complete in one hour? This 
will be my rate.

Now you have two rates, in the form (portion of house) per (hour).  
You want to know the amount of time (in hours) that it will take for 
the two of us to complete one house.  Let's look at a similar problem.

Suppose I know that there are 12 inches in a foot, or 12 inches per 
foot. And I want to know how many feet are in 36 inches. Well, I set 
up an equation that looks like this:

   12 inches
   ---------  x  (some number of feet)  =  36 inches
    1 foot

Well, just as numbers in a fraction cancel with one another, so do 
units. So to solve the above, we do the following:

 1 foot     12 inches                                        1 foot
--------- x --------- x (some number of feet) = 36 inches x ---------
12 inches    1 foot                                         12 inches

which we obtain just by multiplying each side of the equation by 
(1 foot)/(12 inches).  Then the left side simplifies easily so that we 
are left with:

                                        1 foot
   (some number of feet) = 36 inches x ---------
                                       12 inches

Now, on the right side, we can cancel inches with inches, and the 12 
with the 36 to obtain 3. So we have:

   (some number of feet) = 3 x 1 foot

so we have 3 feet for the answer.

Now, how does this apply to your problem? Well, the rates you figured 
out are similar to the feet/inches conversion fraction - each is a 
fraction with different units on the top and bottom. Of course you have 
two rates, one for you and one for me. So you can set up your equation 
as follows:

   (my rate) x (some time) + (your rate) x (some time) = 1 house

Amount of time (in hours) is your variable, but it will be the same for 
each of us, since we are working together to finish the house. So we 
get:

   (my rate + your rate) x (some time) = 1 house

Now, the reason I showed you the conversion example above is that you 
will find that the same sort of units-cancelling will occur here, since 
the rate is in the units (house)/(hours).  

See if you can do the problem now, with these hints, and feel free to 
write back if you need more help. Good luck!

- Doctor Stacey, The Math Forum
Check out our web site! http://mathforum.org/dr.math/