# Math and Science Solutions for Businesses/Compound roportions

Question
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.

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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Math and Science Solutions for Businesses

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#### Randy Patton

##### Expertise

Questions regarding application of mathematical techniques and knowledge of physics and engineering principles to product and services design, optimization, prediction, feasibility and implementation. Examples include sales and product performance projections based on math/physics models in addition to standard regression; practical and cost effective sensor design and component configuration; optimal resource allocation using common tools (eg., MS Office); advanced data analysis techniques and implementation; simulation and "what if" analysis; and innovative applications of remote sensing.

##### Experience

26 years as professional physical scientist and project manager for elite research company providing academic quality basic and applied research for government and defense industry clients (currently retired). Projects I have been involved in include: - Notional sensor performance predictions for detecting underwater phenomena - Designing and testing guidance algorithms for multi-component system - Statistical analysis of ship tracking data and development of anomaly detector - Deployed vibration sensors in Arctic ice floes; analysis of data - Developed and tested ocean optical instrument to measure particles - Field testing of protoype sonar system - Analysis of synthetic aperture radar system data for ocean surface measurements - Redesigned dust shelters for greeters at Burning Man Festival Project management with responsibility for allocation and monitoriing of staff and equipment resources.

Publications
“A Numerical Model for Low-Frequency Equatorial Dynamics” (with Mark A. Cane), J. of Phys. Oceanogr., 14, No. 12, pp. 18531863, December 1984.

Education/Credentials
MIT, MS Physical Oceanography, 1981 UC Berkeley, BS Applied Math, 1976

Past/Present Clients
Am also an Expert in Advanced Math and Oceanography