## Conversions

Often while solving problems it is necessary to change from one set of units to another. For example lengths may be given in miles and the answer requires that the lengths be metric, i.e. kilometers. It is then necessary to know how many kilometers there are in a mile:

1 mile = 1.609 kilometers

If the original units were given in feet and the following are known:

1 mile = 1.609 kilometers

1 mile = 5280 feet

Then two conversions are needed: from feet to miles and then from miles to kilometers.

To carry out conversions then two things are needed:

1. the relationship between the two units
2. the best way to make the conversion.

1. Item 1 is usually found by looking in standard conversion tables. Some are given below, some are in your textbook, and some may be found by looking in a reference book such as the Handbook of Chemistry and Physics, which lists conversions for a large number of units.
2. There are a number of ways of actually making the conversions but the one that is most likely to avoid errors involves making a ratio from the conversion units that equals 1. This ratio can then be used to multiply the original units to achieve the conversion. Since the ratio = 1 this multiplication does not change the item, it just changes the units.

For example:

If the distance between two buildings is 2000 feet, what is the distance in kilometers?

Since 1 mile = 5280 feet

1 = 5280 feet/1 mile

(the ratio 5280 feet/1 mile equals 1 was arrived at by dividing each side of the original conversion equation given above by 1 mile)

1 = 1 mile/5280 feet is also true

(the ratio 1 mile/5280 feet equals 1 was arrived at by dividing each side of the original conversion equation by 5280 feet)

Also since 1 mile = 1.609 kilometers

1 = 1.609 kilometers/1 mile and

1 = 1 mile/1.609 kilometers

2000 feet = ? kilometers. Convert feet to miles by choosing the conversion ratio that will eliminate the feet units: 1mile/5280 feet. This accomplishes the goal since the feet term in the denominator of the ratio cancels the feet term in the numerator of the original number.

2000 feet x 1mile/5280 feet = 2000/5280 miles = .379 miles

Next convert miles to kilometers in the same way. Chose the conversion ratio that will eliminate the miles units (i.e. the ratio that has miles in the denominator) 1.609 kilomters/1 mile = 1

.379 miles x 1.609 kilomters/1 mile = .610 kilometers.

Of course the two steps may be written in one equation:

2000 feet x 1mile/5280 feet x 1.609 kilomters/1 mile = .610 kilometers.

The advantage of carrying out conversions in this way is that you can keep track of what you are doing and so you can be sure that you carried out the conversions correctly.

Some useful conversion equations are given in the tables below. (Unsure of the abbreviations used below? Click here). If you do not find what you want or would like to try other conversions click here

 LENGTH 1 m 100 cm 1 mm 1000 m 1 m 3.281 ft 1 ft 0.3048 m 1 ft 30.48 cm Can you calculate this from the information above? 1 mi. 1.609 km 1 mi. 5280 ft 1 in 2.54 cm 1 ft 12 in 1 nm 10-9 m 1 Å 10-10 m Angstrom unit 1 AU 1.5 x 108 km Astronomical unit

 AREA 1 ha 100 m2 1 mi2 640 ac

 VOLUME 1 L 1000 cm3 1 L 1000 mL 1 mL 1 cm3 1 ft3 7.481 gal

#### MASS

1 kg

1000 g

1 mg

.0001 g

1 oz

.0352 g

1 kg

2.2 lb.

1 lb.

16 oz

1 ton

2000 lb.

1 metric ton (= 1 tonne)

1000 kg