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# Word Problems/Compound Proportions

Question
QUESTION: Hello:

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

Here is a solution:

6300 dollars/(35 students × 24 days) = 6300 dollars/(840 student days) = 7.50 dollars/(student day)

3375 dollars/(25 students) × 1 student day/(7.50 dollars) = 18 days

25 students will be charged \$3375 for 18 days.

Does it make a difference which quantities or units to use as the unit rate?

In the solution, dollars per student per day are used as the unit rate. Could students per dollar per day be used, or days per student per dollar also be used?

ANSWER: The quantities were used the way there were suppose to be.

Since there is a total of \$6,300 for 35 students over 24 days,
it needs to be divided by 35 students to get the amount per student for
over 24 days.

Since we need the amount per day, it should be divided by 24 day.
This cancels days, and the amount left is dollars when we have 1 student over 1 day.

Since this is the amount per day, it needs to be multiplied by 18
to get the amount over 18 days.

Yes, you did the calculations correctly.

To make sure the problem was done the right way, we just need to make sure the units that are not being looked for all cancelled.

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

QUESTION: Hello:

You indicated that the "Quantities were used the way there (sic) suppose(d) to be."  How are they supposed to be used or arranged and why?

Consider this example:

10 men can lay a road 75 Km. long in 5 days. In how many days can 15 men lay a road 45 Km. long?

What is used as the unit rate?

10 men/(75 Km X 5 days), 75 Km/(10 men X 5 days), or 5 days/ 10 men X 75 Km), or something else?

I need to understand what arrangement should be used as the unit rate. I understand how the quantities cancel but I do not understand how to arrange them for the proper unit rate.

We are given 10 men, 75 Km of road, and 5 days of time.
If we think about it logically, doubling the men will half the time.
Doubling the road length will double the time.

The number of days can be seen to be a function of the other two.

Let's takes the function as d(m,k) where m is for the number of me and k is for the length of the road in kilometers.  Note that d(m,r) is the number of days.  It can be seen that when the men are doubled, it would be expected that the days would be cut in half, so let's divide by m.  It can also be seen that as the length of the road is doubled, the time should also be doubled, so lets multiply by r.

That means our function is in days, where the number of days is d(m,r).  It can be seen that
d(m,r) = Kr/m.  Since we have been told when we input m=10 and r=75 that the result is 5,
M can be found.  We just need to say that 5 = M*75/15.  Since 75/15 = 5, we have 5 = 5M,
so M = 1.

That says the equation d(m,r) = r/m.

In how many days can 15 men lay a road 45 Km. long?
This says that m=15 and r=45.  This has d(15,45) = 45/15 = 3.

What this means is before we can make an equation, we need to think about whether the variables go in the numerator or the denominator.  As the affect of doubling each of the units is thought about, think about what that would do to the results.  If it is the same, multiply by them.  If it is inverted, divide by them.  Once the equation has been set up, the constant can be found and different values can be put in.

In problems that arise, it should be made sure that all units are the same.
For example, in distance, they should all be measured in the same units.

I've never said this before, but it still bothers me a little bit.  Now why I think the rest of the word has the advantage in measuring ...

From what I know volume is easily converted between cc's, liters, or kiloliters, distance is easily converted between km, m, cg, and mg, and weight is easily converted between kg, g, cg, and mg.  Note also an in American weights, an object weighs less on the moon, but they still have the same mass, which is what is weighed overseas.  That's because weight is dependent on gravity.

This is one area where the rest of the world has an advantage.  In the USA, weight is in ounces, pounds, or tons.  There are 16 ounces to a pound and 2000 pounds in a ton - most of the time.  If gold is what is being discussed, there are only 12 ounces in a pound.  One pound of other stuff is the same as one pound of gold, but the ounces are different.  In temperature, most of the rest of the world has freezing is 0 and boiling is 100, which makes sense.  In the US, 32 is freezing and 216 is boiling - why?  That's what somebody chose at random.

Now in the USA, we have 12 inches in a foot, 3 feet in a yard, and 5,280 feet in a mile.
That makes a yard 36 inches and a mile 1,760 yards.  Now to get inches in a mile, take
36*1,760 = 63,360, but almost everyone over here doesn't even think about that.  Why don't we have a simple measuring system like the rest of the world?  They've tried to put it in, but it's not there yet.   At least our doctors have weight and volume like your countries do.

Now time, I believe, is still the same around the world, as in 60 seconds in a minute, 60 minutes an hour, and 24 hours in a day.  The transition from one unit to another is crazy as well.  That makes an hour be 3,600 seconds and a day be 864,000 seconds, but most don't care.

What this means is when we tell somebody something is 2.6 miles away, we don't even think about how many feet, yards, or inches are involved.  Over seas (in any direction), if someone says something is 2.6 km away, it is known to be 2,600 meters, 260,000 cm, 2.6 million mm.

Oh well ... life goes on.

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Word Problems

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#### Scott A Wilson

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