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/
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