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 Math Help > Physics > Units Arithmetic

# Unit Conversions: Arithmetic using Physics Units

### "Quantities"

In any real-world calculation involving numbers, the numbers represent some physical concept -- a velocity, perhaps, or force, maybe mass.  This concept has a number, certainly, but it also has a "unit".  Velocity is measured in units of miles per hour, or in meters per second.  Force can be measured in pounds or Newtons.  Mass can be measured in kilograms.  So whenever such a concept is presented, it is always presented as a number and a unit: 60 miles per hour; 18 pounds; 34 kilograms.  I call these "quantities".  It is vitally important to know not just the number, but also the unit, before the quantity can be understood and used in any calculation.

### Two-part Calculations using Quantities

You learned in school that you can simplify fractions by cross-canceling.  For example,

 3 � 2 = 1 4 3 2

You can see that this works great for numbers.  Now you will see that the method works just as well for "quantities", which I said consist of numbers and their units.   Here's an example:

 3 revolutions � 60 seconds = 45 revolutions 4 second 1 minute 1 minute

This example shows that if a wheel is turning 3/4 of a revolution per second, you can multiply it by the number of seconds in a minute to convert the unit to revolutions per minute.  Notice that you're really doing two separate cross-canceling exercises -- one for the numbers, and the other for the units.  In the cross-canceling for the numbers, you reduce 60/4 to 15/1, and multiply this by 3 to get 45/1.  In the cross-canceling for the units, you cancel the seconds that appear in the numerator of the second fraction with the seconds that appear in the denominator of the first fraction.  This is the essence of unit conversion.

### Unit Identities

Look at the example, above, in which revolutions per second were converted to revolutions per minute.  The second factor on the left side of the equation is 60 seconds per minute.  This quantity is called a "unit identity" because its combined numerator, 60 seconds, is identical in meaning to the combined denominator, 1 minute.

### Converting Units using Unit Identities

Suppose you want to convert 60 miles per hour into feet per second.  How can you do this?  The trick will be to use the right Unit Identities.  You'll need to write an equation that looks something like this:

 60 miles � xxx time-unit � xxx distance-unit = xxx feet 1 hour xxx time-unit xxx distance-unit xxx second

As you can see from the unit on the very right-hand side, you will need to convert the time unit from hours to seconds.  So the unit identity you will need to pick will have hours on the top, so they cancel the hours on the bottom of the first fraction.  This unit identity will have seconds on the bottom, so that the final unit will have seconds in the denominator (feet per second).  As you know, 1 hour equals 3600 seconds, so the unit identity will have 1 hour at the top and 3600 seconds in the bottom.

 60 miles � 1 hour � xxx distance-unit = xxx feet 1 hour 3600 seconds xxx distance-unit xxx second

As you can see, the hours in the top of the second fraction cancel the hours in the bottom of the first fraction.  The seconds in the bottom of the first fraction aren't canceled by anything, so we're left with seconds in the denominator, which is good.  Now, turning our attention to the distance units, we need to cancel miles and end up with feet in the numerator.  So the unit identity we are searching for is one that has feet in the numerator and miles in the denominator:

 60 miles � 1 hour � 5280 feet = 88 feet 1 hour 3600 seconds 1 mile 1 second

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The webmaster and author of this Math Help site is Graeme McRae.