- UNITS:
All measurements must be expressed in the proper units.
- UNIT FRACTIONS:
Basic principle: Any fraction that has equal quantities on both top and bottom (in both numerator and denominator) is equal to one.
A fraction that has numerator and denominator equal (and is therefore equal to one) is called a unit fraction.
Another basic principle: You can always multiply or divide by one without changing anything: 1 x A = A = A x 1 = A / 1 = A
Therefore, you always multiply (or divide) by a unit fraction without changing the expression.
- UNIT CONVERSIONS:
You can always pick a unit fraction to convert from bigger to smaller units.
For example: In the metric system, 100 cm = 1 meter
Therefore, 100 cm / 1 meter is a unit fraction, and so is 1 meter / 100 cm
Thus, we can convert from meters to centimeters (and vice versa) by using unit fractions:
| 5.4 cm | = | 5.4 cm x | _ 1 m _
100 cm
| = | _ 5.4 x 1 _ cm x m
100 cm | = 0.054 m
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Now, study the final step carefully, and notice two very important things:
- 1. We do all the arithmetic at the end, in the final step (using a calculator if needed), and
- 2. We cancel (or cross out) all units that occur in both numerator and denominator, and only write the ones than remain.
Here's another example: 1000 g = 1 kg so 1000 g / 1 kg is a unit fraction, and so is 1 kg / 1000 g
So we can convert from grams to kilograms (and vice versa) by using unit fractions:
| 703.1 g | = | 703.1 g x | _ 1 kg _
1000 g
| = | _ 703.1 x 1 _ g x kg
1000 g | = 0.7031 kg
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Notice that we always put the units that we want in the numerator of the unit fraction,
and the units that we want to get rid of in the denominator.
We can also convert between metric units and British units: 1 inch = 2.54 cm so 1 in / 2.54 cm is a unit fraction, and so is 2.54 cm / 1 in
So we can convert from inches to centimeters (and vice versa) by using unit fractions:
| 62 cm | = | 62 cm x | _ 1 in _
2.54 cm
| = | _ 62 x 1 _cm x in
2.54 cm | = 24.4 in
|
Again, notice that we put the units that we want in the numerator of the unit fraction,
and the units that we want to get rid of in the denominator.
Another example: If 1 inch = 2.54 cm, convert 3.30 miles into metric units. Here's how:
| 3.30 miles | = | 3.30 miles x | 5280 ft
1 mile | x
| 12 in
1 ft | x | 2.54 cm
1 in
| = | _ 3.30 x 5280 x 12 x 2.54 _miles ft in cm _
1 mile ft in
| = 531083.52 cm
| | or 531083.52 cm x (1 m / 100 cm) = 5310.83 m
| | or 5310.8352 m x (1 km / 1000 m) = 5.31083 km = 5.31 km to 3 significant figures
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Notice that numerical calculations are all done at once, at the end, and that (miles/miles), (ft/ft),
and (in/in) all cancel in the step leading up to the answer shown in red.
Also notice that rounding off to the correct number of significant figures should be done last of all.
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