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QUESTION: Hello:

Can you fully explain what I need to know in order to determine what is directly and inversely proportional in the example below?

If 195 men working 10 hours a day can finish a job in 20 days, how many men employed can finish the job in 15 days if they work 13 hours a day?

Solution:

(20 x 10 x 195) / (15 X 13) = 200 men

I want to know how to determine what is directly and inversely proportional in order to get the above solution.

I thank you for your reply

ANSWER: Let me show how the solution comes about and then try to answer your direct and inverse proportionality question.

The first thing to recognize is that the "job" is an entity that remains constant and has units of

1 J job = (N men)x(M hours/day)x(D days).

This can be rewritten N = J/(MxD).

Plugging in the numbers for the first crew we get J = (195 x 10 x 20)(men･hours/day･days).

Thus, for the 2nd crew we have N = J/(MxD) = (195x10x20)/(15x13) = 200 men.

It can be seen that the number of men needed for a job is proportional to J (job) and inversely proportional to the hours per day and the number of days.

Let me know if this answers you question.

Randy

---------- FOLLOW-UP ----------

QUESTION: Hello:

I want to thank you for the reply.

Here is what I am trying to understand.

The more men working the fewer days are needed to finish the job more quickly. So, more men fewer days, inverse relationship between men and days. I thought that the same would be true for hours worked. More men working fewer hours needed to work per day to complete the job. the opposite is true fewer men working more time is needed both in days and hours.

So when I set up the calculation I used the inverses: of men to days and men to hours. I would look like this: 13/10 X 20/15 X 195

But this calculation is incorrect. I used the inverses of days and hours. What did I do incorrectly?

I thank you for your follow-up reply.

I think what you say in words is correct:

- more hours worked per day means less men needed for a given job (all else being equal): inverse relationship between men and hours/day

- less days worked means more men needed (all else being equal): inverse relationship between men and days.

We want to use the change in hours/day and the days worked between the 2 crews to calculate the new number of men needed. These changes are expressed as ratios since we are multiplying factors to get the total. Your expression is almost correct; I get

195 x (10/13) x (20/15) = 200.

^ ^

change in change in

hours/day days.

The first ratio in parantheses is less than 1 and reflects the fact that more hours/day will be worked by crew 2 and so, all by itself, would indicate that fewer men would be needed. The 2nd ratio is greater than 1 indicating that fewer work days are planned and so more men will be need to finish the job.

I think what you wanted was a way to multiply the initial number of men by the appropriate ratios to get the new number of men needed for the job. I think this calculation should do it.

Hope this helps.

Randy

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