Unit Multipliers

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Before I delve into this subject, I want to review one basic concept that needs to be understood before learning how to do unit multipliers.

Concept #1: Writing any whole number as a fraction while still retaining its value:
How do we write 4 as a fraction? 4/1 A number can be changed into a fraction and still keep its value by placing 1 as the denominator. How do we write 1 as a fraction? Many ways, 1/1, 2/2, 84/84.

Examples of Concept #1:

What Concept #1 has to do with unit multipliers:

Unit multipliers are fractions that equal 1. The difference between ordinary fractions that equal one and unit multipliers is that unit multipliers require units.

What do you mean by units? In the case of unit multipliers, I am referring to units of measure such as feet, inches, millimeters, centimeters,..... all units of measure, linear (length), liquid, mass, weight,......

Now that the formalities are over with, let's look at a unit multiplier...

Remember how to make a fraction that equals 1? Here are a couple of unit multipliers that equals 1:

It is true! Both of the fractions above equals one. Do you see why it equals 1?

Here's a clue......

12 inches = 1 foot and
1 foot = 12 inches

So do you see why it equals 1?

Let's look closer.....

So the "units" in unit multipliers are important and completely necessary! Do not leave them out!

Here are some more examples of unit multipliers:

I am thinking that you can make your own unit multipliers now! Below are some equivalence tables that you can use to help you make unit multipliers.

Equivalence Table for Units

U.S.

METRIC

1 gallon (gal) = 4 quarts (qt )
1 quart = 2 pints (pt)
1 pint = 2 cups (c)
1 pint = 4 gills
1 cup = 8 ounces (oz)
1 ounce = 2 tablespoons (TBL)
1 tablespoon = 3 teaspoon (tsp)
12 inches (in) = 1 foot (ft)
3 feet (ft) = 1 yard (yd)
40 rods = 1 furlong
8 furlongs = 1 mile (mi)
1 mile = 5280 feet
1 mile = 1760 yards
16 ounces = 1 pound (lb)
2000 pounds = 1 ton

1 liter (L) = 1000 milliliters (mL) [liquid]
and 0.001 L = 1 mL
1 cubic centimeter (cc) = 1 mL [liquid]
10 millimeters (mm) = 1 centimeter (cm)
and 0.1mm = 0.01cm
0.1mm = 0.0001 meter (m)
and 1000 mm = 1 m
100 cm = 1 m
and 0.01 m = 1 cm
1000 m = 1 kilometer (km)
and 0.001km = 1m
1000 gram (g) = 1 kilogram (kg) [mass]

Other and Between....

2.54 cm = 1 in (memorize this one)
1 cm = 0.3937 in
1 mi = 1.6093 km
1 km = 0.6213 mi
1 yd = 0.9144 m
1 m = 39.3701 in
14.59 kg = 1 slug [mass]
3.78 L = 1 gallon [volume]
3600 seconds (s) = 1 hour (hr)

Now that you are familiar with unit multipliers, let's talk about how to use them. You will use unit multipliers to convert the unit of a denominate number while still retaining the value of the denominate number.

What's a denominate number??

denominate number, pronounced dih NAHM uh niht, tells the amount of a quantity by giving the number of units and the kind of units that make up the quantity. A denominate number includes a number, which may be written as a numeral or as a word, and the name or symbol of a unit of measurement. Six meters, 90 pounds, $500 million, 50 miles, 23 DEG>C, and 4 days are all denominate numbers.

Let's look at how to use unit multipliers....

First of all, let's say that I want to know how many inches 5.5 feet are. We understand that we are working with feet and inches.

First, I will change 5.5 feet (a denominate number) into a fraction:

Second, I think about what unit multiplier that I can use to convert feet to inches. I know that 12 inches = 1 foot so the available unit multipliers are:

Which one do I choose? It does make a difference and it is important to choose the right unit multiplier. Fortunately, it's an easy choice.

Choose the unit multiplier that has same unit (in this case, feet) in the denominator place. (?) In other words, because feet is the numerator in the fraction: 5.5 feet/1, you should have feet as the denominator in your unit multiplier.

We must choose as the unit multiplier to convert 5.5 feet into inches.

Remember... the unit multiplier equals one. When you multiply a denominate number and a unit multiplier, the denominate number retains its value because any number that is multiplied by one stays the same. The change will be with which unit the value is expressed.

Here's how it is done...

How about a couple of other examples to set this better in your mind...

Example #1:

Let's convert 243 feet into yards.

1. Change 243 feet into a fraction. The value of 243 feet is still true.

2. Choose the correct unit multiplier. This one is chosen because the feet unit is the denominator.

3. Cancel out. This is really called simplifying, but it "cancels" the feet.

4. Multiply.

5. Simplify.

The denominate number 243 feet equals the denominate number 81 yards.

Example #2:

Convert 629 centimeters into meters.

1. Make 629 centimeters a fraction.

2. Choose a unit converter.
The possibilities...
Both of these unit converters will work. Cm. is the denominator and m. is the numerator. They both equal one. I will choose the first one and you can pick the second one.