Dimensional Analysis and Temperature Conversion

Date: 04/13/2002 at 20:12:24
From: Kevin McGushin
Subject: Can't cross multiply for temperature conversion

Dr. Math,

I am having some confusion on the properties of cross multiplication 
and metric conversion. I understand that you can use cross 
multiplication to find how many centimeters equal a certain number of 
inches - for instance 1 inch = 2.54 cm, so if I want to find how many 
centimeters are in 5 inches, I can just cross multiply 5 x 2.54 to 
find that there are 12.7 cm in 5 inches. 

However I get confused when I apply the same logic when trying to 
convert from Celsius to Farenheit, or vice versa. It does not work the 
same way. For instance if I am trying to convert 95F to Celsius, why 
can't I multiply 95 by 100 and divide by 212 to get 44.8 C? This is 
wrong because 95 degrees F is actually equal to 35C. Why does this not 
work?

Thank you.

Date: 04/13/2002 at 22:44:01
From: Doctor Peterson
Subject: Re: Can't cross multiply for temperature conversion

Hi, Kevin.

I'm not familiar with cross-multiplying for unit conversion, or at 
least with calling it that; my preferred way to explain the conversion 
is "dimensional analysis," in which we include the units in fractions 
(as if they were algebraic variables) and make sure we multiply in a 
way that cancels them out. For example, 5 inches is

               2.54 cm
    5 inches * ------- = 12.7 cm
               1 inch

where we multiply by a fraction equal to one (because 2.54 cm and 1 
inch are the same thing), and "inches" cancels out.

But when we convert Celsius to Fahrenheit, we are not only changing 
units, but also changing the starting point for our scale. For the 
unit, a difference of 180 degrees F (from 32 to 212) is the same as 
100 degrees C (from 0 to 100), so we can multiply by

    180 degF
    --------
    100 degC

But that doesn't deal with the fact that the two scales do not have 
zero at the same place. So after we have adjusted the _size_ of the 
unit this way, we have to fix the _starting point_ by recognizing that 
the Fahrenheit temperature is that many Fahrenheit degrees _above 32_ 
rather than above 0. So the temperature is

             180 degF
    X degC * -------- + 32 degF = (9/5 X + 32) degF
             100 degC

In reverse, given a Fahrenheit temperature, we have to first find how 
far it is from freezing by subtracting 32, and then change the unit:

                         100 degC
    (X degF - 32 degF) * -------- = (X - 32)*5/9 degC
                         180 degF

I hope that helps clarify what is going on here.

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