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Word Problems/Compound Proportions

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QUESTION: The following is an example as to how to solve a compound proportion.  Can you explain what “directly related” indicates and what “inversely related” indicates although this term is not use in this example? Please explain clearly.


Quantity 1----Quantity 2----Quantity 3
----a------------b-------------c
----d------------e-------------?
 

If quantity 1 and quantity 2 are [directly related] and quantity 2 and quantity 3 are also [directly related], then use the following rule:

(a X b)/c = (d X e)/?

Example:
If 195 men working 10 hours a day can finish a job in 20 days. How many men employed are needed to finish the job in 15 days if they work 13 hours a day?

Solution:


Days----Hours-------Men
20--------10--------195
15--------13--------?

20 X 10 X 195 = 15 X 13 X ?

(20 X 10 X 195)/(15 X 13) = ?

What are "directly and inversly related" indicate or mean?

I thank you for your reply.

ANSWER: Directly related means the both go up or both go down at the same time.

The hours worked and the cost are directly related,
for as one increases, the other increases.

The number of workers and the completion time are inversely related.
As the number of workers increase, the completion time decreases.

The numerator is 20 x 10 x 195, and that is 39,000.
The denominator is 15 x 13, and that is 195.

Since 195 x 2 = 390, the answer has 2 with two zeros,
for 39,000 / 195 = 2*39,000 / 390 = 2 * 100 = 200.



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

QUESTION: Hello:

I want to thank you for the reply.

Can you re-explain what you have typed because it is somewhat confusing?

The quantities are men, days, and hours.  You have "cost" and "completion time."
There is no "cost." Is completion time days or hours?

"The hours worked and the cost are directly related,
for as one increases, the other increases.

The number of workers and the completion time are inversely related.
As the number of workers increase, the completion time decreases."

I thank you for your follow-up reply.

Answer
The cost is the amount per hour.
The days are really time in the hours worked per day.
Most people work 8 hours per day, so if you have d days and h hours, the total times is 8d + h.

Rather than trying to restate the old problem, lets look at another one that is slightly simpler.

If making a widget took 96 hours, 1 person would take 12 days, since 12*8 = 96.
If we had 2 people, it would only take 6 days, since 2*6*8 = 96.
It can be seen that people and days are inversely related, for more people mean less days.

If two widgets were made, it would take 2*96 = 192 hours.
Thus, the number of widgets and the time are directly related,
for increasing the number of widgets increases the time needed to make them.

Directly related means as one increases, the other increases.
Directly related means as one decreases, the other decreases.
Basically, directly related means they go in the same direction.

Inversely related means as on increases, the other decreases.
Inversely related would also mean as one decreases, the other increases.
Basically, inversely means they are go in the opposite direction.

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