Applications

Let’s say you’re a forestry technician working for the Washington Department of Natural Resources (DNR). You need to calculate the acres of timber included in a sale. The unit is 15.3 square miles, but you need to omit the area of forestland within the Riparian Management Zone (RMZ). The RMZ buffer width for streams on the site is 67 feet on each side and you have 1.6 miles of stream within the sale unit.

Length multiplied by width will give you the area of the RMZ. What do you need to know to calculate the acres of timber available for harvest in this unit?

An axiom is a truth. In math this refers to a property of numbers. Understanding a few of these can help us understand an easy way to convert between different units.

1. Anything divided by itself is equal to ONE. [latex]\frac{2}{2}=1[/latex]    [latex]\frac{4325}{4325}=1[/latex]    [latex]\frac{apple}{apple}=1[/latex]
2. Multiplying by ONE changes nothing. [latex]5*1=5[/latex]    [latex]4325*1=4325[/latex]    [latex]apple * 1 = apple[/latex]
3. Conversion factors are equal to ONE.  1 foot = 12 inches  \frac{1 foot}{12 inches} = 1
4. In a multiplication, factors that are identical in the numerator and the denominator “cancel” and can be removed from the expression.  [latex]\frac{4}{5}*\frac{7}{4}= \frac{7}{5}[/latex]

So, we can multiply a number by any conversion factor and it is the same as multiplying that number by ONE. Nothing changes. If we arrange the conversion factors so that we have the same units in the numerator as we have in the denominator, these units cancel and can be removed from the expression. You’ll want to see a few of these to get the hang of it!

We could solve unit conversions using proportions, but there is another method that is more versatile, especially when a conversion requires more than one step. This method goes by various names, such as dimensional analysis or the factor label method. The basic idea is to begin with the measurement you know, then multiply it by a conversion ratio that will cancel the units you don’t want and replace it with the units you do want.

It’s okay if you don’t have the conversion ratios memorized; just be sure to have them available.

Let’s walk through two examples to demonstrate the process.

Suppose you’re a fan of Eminem^[1] or the Byrds^[2] and you’re curious about how many feet are in [latex]8[latex] miles. We can start by writing [latex]8 mi[/latex] as a fraction over 1 and then use the conversion ratio [latex]1 mi = 5280 ft[/latex]to cancel the units.

[latex]\frac{8 mi}{1} \cdot\frac{5280 ft}{1 mi} = 42,240 ft[/latex]

Now suppose that you want to convert a measurement from feet to miles. (Maybe you're watching The Twilight Zone episode "Nightmare at 20,000 Feet"^[3] and wondering how many miles high that is.) We'll start by writing [latex][/latex]20000 ft[latex][/latex] as a fraction over 1 and then use the conversion ratio [latex][/latex]1 mi =5,280 ft[latex][/latex] to cancel the units.

\frac{20000 ft}{1} \cdot\frac{1 mi}{5280 ft} = 3.8 mi

As it happens, the first situation became a multiplication problem but the second situation became a division problem. Rather than trying to memorize rules about when you'll multiply versus when you'll divide, just set up the conversion ratio so the units will cancel out and then the locations of the numbers will tell you whether you need to multiply or divide them.