Units Cancellation

Some problems are more complex than the examples we have used, and many do not have patterns or previously developed equations. Equations can be developed for some of these problems, but an alternative approach is units cancellation. Problems of this type will usually involve several quantities. All of these quantities, except n, will have a unit such as feet, pounds, gallons, and so on. Units cancellation follows two mathematical principles: (1) the units of measure associated with the numbers (feet, gallons, minutes, etc.) follow the same mathematical rules as the numbers; (2) the units of the numbers behave according to the rules of fractions. For example:

2 x 2 = 4 or 22 With units of feet the same equation is:

2ft x 2 ft = 2 x 2 and ft x ft or 4 ft2 To review the rules of fractions study the following example:

In this example, the 4's in the numerator and denominator cancel out (4/4 = 1).

When the units of measure are included, they behave in the same way:

4 hr 5 day 20 day day

In this example, the 4's and the units associated with them cancel out. The uncancelled units become the units for the answer. The following example shows another variation of this principle (where gal = gallon and hr = hour):

In this example, the unit of hour in the numerator and denominator cancel each other, leaving the units of the answer in gallons.

Problem: What is the weight (lb) of one pint of water?

Solution: If a scale and a one-pint measure were available, it would be a simple task to weight one pint of water. An alternative is to use the conversion factors found in a table of weights and measures (Appendix I) and units cancellation.

Note, in this example two types of measure are used, volume and weight. The real nature of the problem is to find the conversion value(s) that will convert from volume (pints) to weight (pounds).

To begin, refer to Appendix I and identify conversion factors that use both volume and weight. You should find that 1 cubic foot contains 7.48 gallons, and 1 gallon contains 8 pints. This is a start, but you need something more. If you also know that water weighs 62.4 lb per cubic foot, the problem can be solved with (lb = pounds, gal = gallons, pt = pints, and ft3 = cubic feet):

The units of pints, gallons, and cubic feet all cancel each other leaving the answer in the desired units of pounds and pints. This example illustrates several principles of units cancellation.

• It is very important to begin by writing down the correct units for the answer.

• Begin entering the values and their units. The first value entered should have one of the desired units in the correct position (numerator or denominator), even if it is a units conversion value from Appendix I or another source. Starting with one of the units of measure in the correct position will eliminate the possibility of having the problem inverted.

• Enter a value that will cancel out the unwanted units, if any, in the first value entered.

• Continue to add variables with the appropriate units until the only units that remain are the units of the answer.

• If all of the units cancel except those that are desired for the answer, and the units are in the correct position, then the only possible mistake is a math error.

The process of units cancellation is also useful for problems requiring the development of a new unit. For example, a very common quantity in agriculture is power. Power can have different units, depending on whether it is electrical or mechanical. The units of mechanical power are ft • lb per minute. The solution to a problem in which a 24-oz weight was moved 15 ft in 5 sec would look like this (with oz = ounces, sec = seconds, lb = pounds, ft = feet):

(Note that lb • ft is a compound unit, not feet minus pounds or feet times pounds.) This same process will work just as well for problems with units that are more complex and more variables.

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