Marveling At The Historical

Math Oldies But Goodies

  • About This Blog

    This blog is mostly about math procedures in textbooks dated from about 1825-1900. I’m writing about them because some of the procedures are exquisite and much more powerful, and simpler, than some of the procedures in current text books. Really!

    I update this blog as frequently as possible ... every 2-3 days. And, if you are a lover of old texts and unique procedures, you might want to talk to me about them, at markdotmath@gmail.com. I’m not an antiquarian; the books I have are dusty, musty, brown-paged scribbled-in texts written by authors with insights into how math works. Unfortunately, most of their procedures have vanished. They’ve been overcome by more traditional perspectives, but you have to realize that at that time, they were teaching the traditional methods.

Danny, Mike and Mary Work Together

Posted by mark schwartz on April 9, 2016

Introduction:

A common problem presented in Algebra texts states that one person does a job in so many hours and that a second person does it in so many hours. The student is asked to find how long the job would take if the two people worked together. This type of problem can also give one person’s time and then the time it takes for both people to do the job and the student is to find the time for the second person working alone. When we were working on these type of problems, the fraction work was distressing to many students and they were interested in finding out if there was an alternative method. As we played, what follows is what came out.

The Story:

For example, Danny takes 4 hours to do a job alone. His brother Mike can do the job alone in 6 hours. If they work together, assuming no gain or loss of efficiency, how long will it take them to do the job?

Traditionally,    1/4 + 1/6 = 1/x      or       x/4 + x/6 = 1

In both cases, the strategy is to set up an equation with fractions and solve for ‘x’, the time together. Regardless of the actual times worked, the formula is the same. Given that, if Algebra is applied to Algebra, we can state the following:

1/a + 1/b = 1/x       or           x/a + x/b = 1

In both cases, the class solved for x and got: x = ab/(a + b). In this case “x” is the time to do the job together.

Call this Equation 1. For the problem stated above, the shared time to do the job would be:

x =  ab/(a + b) = (4⦁6)/(4 + 6) =   = 2 hrs. 24 min

If given the time of one individual and the shared time, Equation 1 can be used to find the time of the other individual. For example: If Mary and Carla can do a job together in 1 hour and 12 minutes, how long would it take Mary to do it alone, if Carla can do it alone in 2 hours? 1 hour and 12 minutes is 1 & 1/5 or 6/5 hours working together. Using Equation 1, the problem can be seen as:

6/5 = 2m/(2 + m) … the answer being that Mary would take 3 hours working alone.

This is particularly useful, since students seem to struggle with problems having fractions in the numerator or denominator (or both) of a fraction (somehow the fraction in Equation 1 wasn’t as daunting as either of the traditional setups with fractions). Equation 1 was worked out with students, so that they had ownership of it and were aware of the value of applying Algebra to Algebra.

This discussion and development of Equation 1, led one student to ask “What if there are 3 people?” A quick reply from another student was: “Just take any two and take that answer and do it with the third person”. I asked “Would that work?” So we tried it with the above Danny and Mike problem and added Mary, who could do the job alone in 5 hours. So, given that Danny and Mike can do it together in 2 hrs. 24 min. (or in hours, 2 and 2/5 hours), the problem would set up as:

x = ((2 & 2/5)⦁5) ̷ ((2 & 2/5) + 5) = 1hr 37 min. I then asked: Is this right? I also pointed out that this is a somewhat messy problem as well.

Then someone asked “If there’s a formula for 2 people, isn’t there a formula for 3 people?” We talked about this a while and then I pointed out that we had solved an equation for 2 people, so how can that act as a model for 3 people? As a class they agreed that the following equation would work for 3 people.

1 ̷ a + 1 ̷ b + 1 ̷ c = 1 ̷ x

The students then solved this and the result was (abc)/(ab + ac + bc). Putting Danny’s, Mike’s and Jane’s times in this formula gave the same answer, 1hr 37 min.

The class was having fun with this and wanted to extend this to 4 and 5 people, but we stopped at 3. One student proposed that it was unlikely 4 people would work together easily and it would probably take longer than any one of them! I gave the 4 and 5 person problem as a bonus assignment and pointed out the pattern in the denominator, suggesting that it could be done by pattern as well as solving equation 1 extended to 4 and 5. Many, not all, received bonus points for deriving and demonstrating the formula for 4 and/or 5 people working together.

For an instructor, this was nirvana. For the students, it was an insightful exercise in how to use Algebra to establish a relationship and then extend it.

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One Response to “Danny, Mike and Mary Work Together”

  1. sarablack said

    Reblogged this on Sarablack's Blog and commented:
    How to enjoy math, the past, and today. Einstein was right – time is relative.

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