Equations from Word Problems, and Units

Date: 05/19/2003 at 10:46:20
From: Colin
Subject: Number placement in fraction and decimal word problems

In a fraction or decimal word problem, how do I know where to place 
the numbers from the sentence when I'm writing the equation? Example:
In 2 1/4 hours the temperature dropped 13 1/9 degrees. How many 
degrees did the temperature drop per hour?

I understand how to pick the operation, just not where to place the 
numbers. Thank you.

Date: 05/19/2003 at 12:17:05
From: Doctor Peterson
Subject: Re: Number placement in fraction and decimal word problems

Hi, Colin.

It would be more helpful if you had shown us what operation you expect 
to use, and where you think the numbers should be placed in your 
expression; that would tell me where you are having trouble. I myself 
can't separate the two aspects; an operation is meaningless without 
numbers to operate on. Let's instead think about what you are being 
asked to find, and what it means in terms of the quantities you are 
given.

You are given

  the total time, 2 1/4 hours
  the total drop in temperature, 13 1/9 degrees

You want to find

  the rate of temperature drop, in degrees per hour.

That is, in each hour (on the average), how many degrees did it drop?

One way to solve this is to think as if the degrees were objects you 
could count. If I get a certain number of apples every hour, then the 
total number of apples I get in a certain number of hours is found by 
multiplying. Do you see that? If I get 5 apples per hour, for 3 
hours, then I have 3 piles of 5 apples for a total of 3*5 = 15 apples. 
Then to reverse the problem, we have to divide: the number of apples 
per pile (or per hour) is the total number of apples divided by the 
number of piles (or hours). Thinking in this way can make it clearer 
what you want to divide.

You can just think of this as a rate problem, using the formula

  d = rt  [distance = rate times time]

In this case, the "distance" is not a physical distance, but a change 
in temperature; but the same equation is valid. Solving for r, you 
have

  r = d/t

So the rate is found by dividing the total "distance" traveled by the 
time it took.

You can also think in terms of units; you have to do so at some point 
anyway. The units of "degrees per hour" can be written as a fraction:

  deg
  ---
   hr

So we want to divide a number of degrees by a number of hours to get 
the number of degrees per hour.

If you had used the rate equation, you would have divided degrees by 
hours, so you would see that the answer is in degrees per hour. If the 
problem had asked you for, say, degrees per minute, you would have to 
convert. And if you had accidentally divided the time by the 
temperature change, you would find that your units were hours per 
degree, and you would know you were wrong. That's why I say you have 
to check the units no matter how you do it.

So there are three different ways to approach this sort of question, 
all of which tell you not only to divide, but what to divide by what.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/