Unit Conversion: *The Science of mixing apples & oranges*

Before you begin working on mathematical comparisons and perspectives of your own there are a few things to know. All of these things are essential to a good understanding of science, and will make your work to make your own comparisons possible, and much easier.

First, the units of measurement used (kilograms, litres, meters, feet, Joules, and so on) must be treated *algebraically*. This means you must always keep track of units, and that they can never just disappear from an equation. Treat them as an “x” in an algebra equation; they must be multiplied, divided, and raised to powers. For instance:

(x)(x) = x^{2 } x^{5} ÷ x^{2} = x^{3 } (x^{3})^{2} = x^{6}

Earlier we talked about areas, which is simply a width multiplied by a length (π is involved when working with circular areas). Associated with each measurement is a unit, for instance “meters”. If a living room in a house has a floor 20 m by 30 m, we find the area by:

20 m × 30 m = 600 m^{2}

Notice that it is not meters (m) anymore but meters squared (m^{2}). Let’s look at this closely:

20m × 30m = (20 × 30) × ( m × m ) = (600) × (m^{2}) = 600 m^{2}

Notice we treated meters just as if it were an “x” or a “y” in an algebraic equation.

Second is a rather simple but subtle notion, and this is that the ultimate objective of division is to reduce the denominator to 1. For instance, you are told someone goes 67 and one half km in one and one half hours. A bit confusing isn’t it? We really have no comparison to make because all our rates of speed are given as the number of kilometers someone goes in 1 hour, not 1.5 hours.

And this is exactly what division does:

67.5 km ÷ 1.5 h = 45 km ÷ 1 h = 45 km/h

The answer, 45 kph, is much easier to understand than 67.5 km/1.5 hours.

Also notice that in the answer the “1” is understood, or assumed, in the denominator, and that we didn’t just whisk away the “hour” (h) abbreviation. This is because we must treat the “h” as an algebraic variable, and because if there is no number present in front of a variable a “1” is assumed. We don't need to write our answer as: 45 km/(**1**)h

A few more examples: we are told that a crowd has 50,000 people and 100,000 hands. How many hands per person?

100,000 hands ÷ 50,000 people = 2 hands/person

This is an obvious answer, but sometimes it pays to look at the opposite division (or, as it is called, the reciprocal) for a new perspective on things:

50,000 people ÷ 100,000 hands = 0.5 person/hand.

So we find out there is one half a person for each hand.

These simple principles can reduce the most confusing data to something understandable. We are told that our friends went 1069.35 kilometers in 14.258 hours. How fast were they going?

Division tells us quickly:

1069.35 km ÷ 14.258 h = 75 km/h.

Or, on the other hand, using the reciprocal division, they took 0.0133 h/km.

Third, and terribly important is the conversion of units. We would be lost in the swamp if we couldn’t convert one set of units to another. For instance, you need an answer in meters but you are given data that is in feet.

Or, you need an answer in liters but the data is in gallons. So we need an easy way to convert from feet to meters, or gallons to liters, or from anything to anything else for that matter.

In the conversion of units you only need to know two basic things: when you multiply something by 1 you don’t change the number. For example:

250 × 1 = 250

35.7865 × 1 = 35.7865

1,054,003 × 1 = 1,054,003

At least that is easy enough. The other thing to remember is that any number divided by itself, __ or by an equivalent__, is equal to 1:

250 ÷ 250 = 1

35.7865 ÷ 35.7865 = 1

1,054,003 ÷ 1,054,003 = 1

Also (wait for it):

3.28084 ÷ 1 = 1

1 ÷ 0.22702 = 1

1 ÷ 1000 = 1

What? You might disagree with those last three equations but now let’s write them down with their __ units__ attached:

3.28084 ft ÷ 1 m = 1.

There are 3.28084 feet in 1 meter which is equivalent to: 1 m ÷ 1 m = 1.

1 L ÷ 0.22702 gal = 1.

That is, 0.22702 gallon is equivalent to 1 liter so: 1 L ÷ 1 L = 1

1 m ÷ 1000 mm = 1.

Every meter contains 1000 millimeters, so: 1 m ÷ 1 m = 1

Isn’t that clever? The units make all the difference. If we set up, or design, an equation using one of these equivalents we will actually be *multiplying by 1*. If we then watch our units and cancel them algebraically we will have converted a measurement from one form of units to another form of units.

An example. We will convert 24 feet into the equivalent number of meters. We will need the equivalent between feet and meters so that we can construct a term that is equal to one. 1 foot equals 0.3048 meters.

So we can write either ^{1 ft}/_{0.3048 m} or ^{0.3048 m}/_{1 ft} - both of these will be equivalent to one. Which one should we use? We will use the one that will work to algebraically cancel the original feet:

24 ft × ^{0.3048 m} / _{1 ft} =

24 ft × ^{0.3048 m} / _{1 ft} =

24 × 0.3048 m =

7.3152 m

ft canceled out and only m remains. And so there is the answer in meters. Were we to start in meters and convert to feet we would use the other foot-meter equivalent thusly:

7.3152 m × ^{1 ft} / _{0.3048 m} =

7.3152 m × ^{1 ft} / _{0.3048 m} =

( 7.3152 × 1 ft ) / _{0.3048 } =

7.3152 ft / _{0.3048 } =

24 ft

Above we found that our friends went 0.0133 hours/kilometer. This means it took them 0.0133 hours to go 1 kilometer. But who knows offhand how long 0.0133 hours is? Let’s convert it to a more reasonable answer in seconds:

0.0133 h × ^{60 min} / _{1 h} × ^{60 sec} / _{1 min} =

0.0133 h × ^{60 min} / _{1 h} × ^{60 sec} / _{1 min} =

0.0133 × 60 × 60 sec =

47.88 sec

So it takes our friends a bit less than 48 seconds to go 1 kilometer. From here on we can convert anything to anything if we know a pair of equivalent units we can use. Should you be interested you have for your use many tables of equivalents in the Reference section. Others can be found in reference books or on the internet. Simply construct the term you need to cancel out the units you don’t want and generate the units you do want. (Or, easy way, get a conversion app or conversion calculator! :-o)

One more example: Let’s convert 100 miles per hour into meters per second using a variety of “1” equivalents:

^{100 mi} / _{h} × ^{1 h} / _{60 min} × ^{1 min} / _{60 sec} × ^{5280 ft} / _{1 mi} × ^{0.3048 m} / _{1 ft} =

^{100 mi} / _{h} × ^{1 h} / _{60 min} × ^{1 min} / _{60 sec} × ^{5280 ft} / _{1 mi} × ^{0.3048 m} / _{1 ft} =

( 100 × 5280 × 0.3048 m ) / ( 60 × 60 sec ) =

44.704 m / sec

One more trick: How do we convert areas and volumes? This is also simple, areas are based on squares and volumes are based on cubes. When converting these units be sure to square, or cube, __ the associated number__ as well as the unit.

For example: given a volume of 864 cubic inches, convert it to cubic meters. This time we will convert inches to centimeters and then centimeters to meters. Remember it doesn’t matter what equivalence you use as long as you end up with the units you want.

First we’ll need to convert the inches to centimeters and that equivalence is 2.54 centimeters in one inch, but this equivalence will need to be cubed:

(2.54 cm)^{3} / (1 in)^{3}=

(2.54)^{3 }(cm)^{3} / (1)^{3} (in)^{3} =

16.3871 cm^{3} / in^{3}

Let’s solve the problem using this method, I'll skip canceling for you, inspections shows that in^{3} and cm^{3} will cancel leaving us with m^{3}:

864 in^{3} × (2.54 cm)^{3} / (1 in)^{3} × (1 m)^{3} / (100 cm)^{3} =

864 in^{3} × (2.54)^{3} (cm)^{3} / (1)^{3} (in)^{3} × (1)^{3} (m)^{3} / (100)^{3} (cm)^{3} =

( 864 × 16.3871 × m^{3} ) / 1,000,000 =

0.0142 m^{3}

A bit of practice, *and checking your answers*, will soon leave you an expert unit converter.