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Combining Rates of Work, Revealing Constants of Word Problems

Date: 11/13/2011 at 12:42:11
From: Kenneth
Subject: Trying to Solve Without Using an Indirect Proportion

I saw this question in an old textbook on arithmetic:

   If 15 men can do a piece of work in 7 days, in how many days can 21 men
   do the same work?

Now, the authors did not explain what mathematical logic is used to
determine the answer, and they explain proportions in another chapter
later in the book -- so it was not their intention to use proportional
reasoning here. But how would you determine the answer without using an
indirect proportion?

I cannot think of a logical, mathematical approach that does not in some
way require setting up this proportion:

     15 men x 7 days           
   -------------------  =   
         21 men                      


            5 x 7 days
          ------------- = 5 days
              7

But I just resorted to trial and error to assign 21 men as the
denominator, and the product of 15 men and 7 days as the numerator. No
other relationship used the information in a way that made the units work
out and still led to the common sense conclusion that -- with more 
labor -- the job must take less than 7 days. 

My method does not use adequate mathematical thinking. There must be a
better one that still does not use an indirect proportion, but does lead
to the solution, above. What do you suggest?

I thank you for your reply and explanations.



Date: 11/13/2011 at 14:35:22
From: Doctor Jerry
Subject: Re: Trying to Solve Without Using an Indirect Proportion

Hello Kenneth,

Thanks for writing to Dr. Math.

   If 15 men can do a piece of work in 7 days, in how many days can 21 men
   do the same work?

If 15 men can do a piece of work in 7 days, then it takes 15*7 man-days to
do the job.

If you have 21 men, then you would need x days to produce 15*7 man-days.
So,

   21*x = 15*7
      x = 15*7/21
        = 5

Please feel free to write back -- using the URLs at the bottom of this
message -- if you have questions relative to my comments.

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 11/13/2011 at 19:32:36
From: Kenneth
Subject: Thank you (Trying to Solve Without Using an Indirect Proportion)

Thank you, Doctor Jerry, for the reply and your explanation!



Date: 11/21/2011 at 11:47:52
From: Kenneth
Subject: Trying to Solve Without Using an Indirect Proportion

Hello Doctor Jerry:

I want to thank you for the reply and answer to my first question!

Can the answer to the following be determined in the same way that the
answer to my first question was determined?

   Four carpenters can build eight houses in 10 days.
   Two carpenters can build how many houses in 15 days?

I set up these proportions:

   4/2 = 8/? = 10/15

Here,

    4/2   represents the carpenter ratio
    8/?   represents the house ratio
   10/15  represents the day ratio.

I thank you for your reply and assistance!



Date: 11/21/2011 at 12:10:46
From: Doctor Jerry
Subject: Re: Trying to Solve Without Using an Indirect Proportion

Hello Kenneth,

Thanks for writing to Dr. Math.

Ever since I took 9th grade algebra, these "combining rates of work"
problems have always bugged me: I can solve them, but the reason I have
relied on has always seemed a bit ad hoc -- similar to what you wrote
about earlier, with your "trial and error" method guided by common sense
and dimensional analysis.

Your questions have moved me to think about this kind of problem from a
slightly different viewpoint.

Implicit in these problems is an assumption that there is a "direct
proportion" involved. Specifically, the number H of houses that c
carpenters can build in d days is ...

   H = K*c*d,

... where K is some constant, at the moment unknown.

The problems provide information that will determine K.

In the specific problem you raise, we know that

   8 = K*4*10

So, 

   K = 8/40 
     = 1/5

We now can write

   H = (1/5)*c*d

We were asked how many houses can two carpenters build in 15 days. So,

   H = (1/5)*2*15 
     = 6

This seems more straight-forward to me.

Please feel free to write back -- using the URLs at the bottom of this
message -- if you have questions relative to my comments.

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 11/22/2011 at 08:16:50
From: Kenneth
Subject: Thank you (Trying to Solve Without Using an Indirect Proportion)

Thanks, Doctor Jerry, for the follow-up reply.
Associated Topics:
Middle School Ratio and Proportion
Middle School Word Problems

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