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What is the mathematical reason that letters and units cancel from multiplication?

For example, 1/2x * 4x = 2 and 1/(2 ounces) * 4 ounces = 2

The x's and the ounces cancel. What is the reason for the cancellation or why does this occur?

I thank you for your reply.

The reason for this is that it tells you what the measure is in.  I really got a grasp of this concept when I took a class where we had to do measurements.  The answer could be in several different numbers depending on the units used.  The main reason to cancel terms is to make the answer as simple as possible.

For example, if it's 12 feet to the front door, it needs the feet to designate what the measurement is.  Since there is 1 yard / 3 feet, we can multiply 12 feet x 1 yard / 3 feet.
Note that we have feet / feet, so they cancel, leaving the answer in yards.  Since 12 / 3 = 4,
it could also be said that the door is 4 yards away.

If we wanted to know the number of inches, we would multiply 12 feet by 12 inches / foot.
The feet/foot cancel, leaving us with 12 * 12 = 144.  Thus, the front door is 144 inches away.

If we wanted metric units, there is a 1 meter / 39.372 inches, so we would take
144 inches by 1 metere / 39.372 inches.  Thus, 144 inches is found to be 3.66 meters.

The main reason that cancellation occurs is that we need to know what the units of the answer are.  If someone asked how far away the door was, I could tell them it was both 4, 12, 144, and 3.66 units away, but this would tell them nothing.  Perhaps if the recognized 12 is 3*4, they might deduce the 12 was in feet and the 4 in yards, but there's no guarantee.

Perhaps the weight of something was 100.  That is a nice number, but what are the units?
If it is 100 ounces, that is a little over 6 pounds.  If it is 100 pounds, that is fairly heavy.  If it is 100 tons, that would require a massive piece of equipment to even have a hope of moving it.

Another example as to why they cancel is it tells us if the units were computed correctly.
For example, if we have something weighing 100 pounds, and we know that density is 5 pounds per cubic foot, we could determine how many cubic feet were involved.  It can be seen that we need to divide the 100 pounds by 5 pounds per cubic foot giving 20 cubic feet.  This is since we are dividing by what has cubic feet in the denominator which puts it in the numerator.

In other words, we take 100 pounds/(5 pounds / cubic foot), giving us 20 cubic feet.

At the store, this is useful in computing prices.  If something is 2$ a pound, and we but 6 pounds, the priced is the (2$/1 pound)(6 pounds) = 12$.

I hope the examples I gave you give you a better understanding of what units are used for.

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Scott A Wilson


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