Unit conversions for mere mortals

So, you are taking a college science course. Maybe it is physics, maybe it is chemistry, maybe its a lab. Either way, you always end up with these problems that involve unit conversions. You think you have the hang of it, but sometimes you make some mistakes. Here is my explanation for converting units.

Convert units? Me? Why? I have google. Yes, that is true, google (for the most part) does an excellent job at unit conversions. But....I doubt your instructor will let you use google on your test. Don't you think you should have a good idea of how to do it? Don't worry. Unit conversion only involves 1 thing.

Unit conversion is multiplication by 1

Yes, really. The key thing to realize is that units are an important part of a number and we don't want to change what that number represents, just its units. Take for instance the number 5. Suppose I do the following:


What would I get? Let me just work it out.


It is the same thing as what I started with. You might say "duh - you multiplied by 2/2 which is one". And you would be correct. What if I do the following:


Well, that will still be equal to 5 because 6/3 is still 2. In this case, I am still multiplying 5 by 1. If you are ok with this, then you are ready for a real unit conversion.

First, suppose I measure the length of my desk and I get:


Note the the units are important. If I measured the desk and got a value of just 55, that is meaningless. 55 what? 55 chickens? 55 gobstoppers? 55 golden gate bridges? Nonetheless, suppose I want the length of the desk in inches instead of cm? I only need to multiply by "1" in this case, my "1" will be created from a a fraction with 0.394 inches and 1 cm since 1 cm = 0.394 inches. That will give:


Notice that the cm canceled since that unit was on the top and the bottom. Really, you can consider the unit to be like a variable (in essence it is). But HEY! you may say - I have been doing this anyway. Why can't I just say "to convert cm to inches, multiply by 0.394". Yes, this is what happens, but not how it happens. With this way, it is easier to do all sorts of conversions. All you need to know is one relationship (like 1 cm = 0.394 inches - OR - 2.54 cm = 1 inch). Let me do a couple of useful examples:

Convert 3 feet per minute to meters per second:


You see, it is possible to do a series of conversions in a row. The key thing to remember is to multiply by fractions that have equivalent things in the top and bottom and make the units cancel (notice I multiplied by 0.305 m over 1 ft so the ft cancel).

Next example - (this one is tricky for many students): Convert 345 cm2 into m2:


I did this one wrong (sort of wrong) on purpose. This is similar to the mistake many students make. They say there is 1 m2 in 100 cm2. This just is not the case. Here is a simulation of that:


This is a square that is 10 smaller squares by 10 smaller squares. Although there are 10 small squares in one side, there are 100 total small squares in this big square. So, for the above problem, it really should be:


Last example: How fast is 20 furlongs per fortnight in meters per second? Solution: what in the world is a forlong? I know a fortnight is two weeks. Type the following in google:

20 furlongs per fortnight in m/s

Yes, google calculator is pretty awesome. Here are some other fun google conversions to try:

  • the mass of the sun in slugs
  • 2 kg in solar masses
  • 3 cm^3 in ft^2*m

Does that last one even make sense? Sure, I will do this one:


The number from google is slightly different (check it out) perhaps due to rounding.

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If all information resides in area rather than volume, local conversions may not be diagnostic of global big stuff.

4 - 10 = 9 - 15
Add 25/4 to both sides,
4 - 10 + 25/4 = 9 - 15 + 25/4
Write sides as complete squares,
(2 - 5/2)^2 = (3 - 5/2)^2
Take the square root of both sides
2 - 5/2 = 3 - 5/2,
add 5/2 to both sides