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QUESTION: Hello:

How would you solve the following?

If 175 men can dig a canal 3150 meters long in 36 days, how many men are required to dig a canal 3900 meters long in 24 days.

I thank you for your solution.

ANSWER: Hello,

Since work = (rate of work) x (time), we just need to see how rate of work relates to
the number of men.

175 men can dig a canal 3150 meters long in 36 days can also be expressed as...
1 man can dig a canal 3150/175 meters long in 36 days, which simplifies to
1 man can dig a canal 18 meters long in 36 days... which can also be expressed as
1 man can dig a canal 18/36 meters long in 1 day, which simplifies to the easy
rate that 1 man can dig 0.5 meter per day.

Now let x=the number of men needed (assuming they all work at the same 0.5 meter/day rate)
to dig a 3900 meter canal in 24 days.  Hence, the equation we need to solve is:
x(0.5 meter/day)(24 days)=3900 ==> 12x=3900 ==> x=325

Thus, 325 men are needed.



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

QUESTION: Hello:

I want to thank you for your reply and solution!

I am still somewhat confused.  How are rates determined as you determined for my first question?

For example, how are the rates determined for the following?

1. The dorm charges $6300 for 35 students for 24 days.  In how many days will the dorm charges be $3375 for 25 students?

2. 24 men working at 8 hours per day can do a piece of work in 15 days.  In how many days can 20 men working 9 hours per day do the same work?

You do not need to determine the answers just explain how the simplified rates are found.

I thank you for your follow-up reply.

ANSWER: Hello again Kenneth,

There are different rates that can be determined.

For 1, we can calculate the charge for all 35 students per day ($6300/24 days = $262.50 per day), or
the charge per student for 24 days ($6300/35 students = $180 per student), or the charge per student
per day ($6300/35 students/24 days = $7.50 per student per day).

Similarly, for 2, we can calculate the rate for all 24 men per day or per hour, or we can get the
rate per man per day or per hour.

See?

A. Mantell


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

QUESTION: Hello:

I am still having some difficulty with this type of solution.

Instead of determining that 1 man can dig 0.5 meter per day, use 175/3150 men can dig a canal 3150/3150 meters long in 36 days.  This would be 1/18 man can dig a canal 1 meter long in 36 days and this becomes (1/18)/36 of a man can dig a canal 1 meter long in 1 day. Now the rate is 1/648 of a man can dig a canal 1 meter long in 1 day.  If my arithmetic is correct!

How would I determine the unknown using this rate?  I am having some difficulty establishing a calculation for this rate.

Answer
Hello,

Rates are determined by division.  For example, if one travels 100 miles in hours, their
rate is (100 miles)/(2 hours) = 50 miles/hour.

Similarly, if 5 men can dig 100 meters in 2 days, we could calculate:
1. (100 meters)/(5 men) = 20 meters/man every 2 days, or
2. (100 meters)/(2 days) = 50 meters/day for 5 men working together, or
3. (100 meters)/(5 men)/(2 days) = 10 meters/day/man.

See now?

A. Mantell

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Abe Mantell

Expertise

Hello, I am a college professor of mathematics and regularly teach all levels from elementary mathematics through differential equations, and would be happy to assist anyone with such questions!

Experience

Over 15 years teaching at the college level.

Organizations
NCTM, NYSMATYC, AMATYC, MAA, NYSUT, AFT.

Education/Credentials
B.S. in Mathematics from Rensselaer Polytechnic Institute
M.S. (and A.B.D.) in Applied Mathematics from SUNY @ Stony Brook

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