There are so many units that you can use to express results that you need to become proficient at converting from one to another. Fortunately, there is an easy way to do this and it works every time with no exceptions! You simply multiply by one. (The only question is: which one?)

Since 12 in = 1 ft, you can get a one by dividing both sides of the equation by 1 ft, giving 12 in

1 ft = 1.

(Notice that this doesn’t say that 12 = 1.) You use this or any similar factor by noticing that if you multiply anything by one, you don’t change it. It’s easy to see with an example; each factor in parentheses equals one.

1 yr = 1 yr · 365.24 days 1 yr · 24 hr 1 day · 60 min 1 hr · 60 sec 1 min = 1 /yr · 365.24 /day 1 /yr · 24 /hr 1day/ · 60 /min 1 /hr · 60 sec 1min/

= 365.24 · 24 · 60 · 60 · sec = 31556736 sec = 3.1557 × 107sec,

where there are only 5 significant figures in the result, because that’s the limit on the accuracy of the input data for the number of days in a year. (This is about π × 107seconds — easier to remember.) Another example:

1 mi 1 hr = 1 mi 1 hr 5280 ft 1 mi 1 hr 60 min 1 min 60 sec = 1 /mi 1 /hr 5280 ft 1 /mi 1 /hr 60min/ 1 /min 60 sec = 5280 3600 ft sec. = 1.47 ft/sec

For an example of what can go wrong, what is one mile in inches?

1 mi = 1 mi · 5280 ft 1 mi 1 ft 12 in = 1 /mi · 5280 ft 1 /mi 1 ft 12 in = 5280 12 ft2 in.

This isn’t wrong , but it’s pretty silly, and isn’t responsive to the original question of con-verting miles to inches. The key point to notice is that I kept track of the units honestly and I didn’t assume that I’ll never make a mistake. A common error (the common error) in doing these conversions is to write down the conversion factors and then to assume that they do what you want them to. If you skip the steps shown here, where you explicitly put in the units and cancel them one pair at a time, you have about a 50-50 chance of messing it up. After all, if 12 in/1 ft = 1, then so does 1 ft/12 in = 1, and you can multiply anything by one without changing it.

mi ft 1 mi 3 ft

Don’t forget that each factor in a conversion must be done; if a unit is squared, its conversion factor enters twice. Convert one square mile to square yards:

1 mi2 5280 ft
1 mi
2_{ 1 yard}
3 ft
2
= 3 097 600 yard2.

(I like the metric system more and more.)

Convert 10 cm1/2 using meters instead of centimeters.

10 cm1/2· m
cm
1/2
then 10 cm1/2·
1 m
100 cm
1/2
= √10
100m
1/2 _{= 1 m}1/2_{.}

For a particularly odd unit that I once encountered, convert 10 cm2/3gm−1/3sec2 to meters and kilograms. First, just get the units in place:

10 cm2/3gm−1/3sec2· m
cm
2/3
·
kg
gm
−1/3
THEN
10 cm2/3gm−1/3sec2·
1 m
100 cm
2/3
·
1 kg
103_{gm}
−1/3
= 10
1002/3· 10−1m
2/3_{kg}−1/3_{sec}2
= 100
10 · 101/3m
2/3_{kg}−1/3_{sec}2
= 10
101/3 m
2/3_{kg}−1/3_{sec}2

Practice Set 1

[1] 2.54 cm = 1 in. What is 1 meter in inches?

[2] g = 9.80 m/s2. Convert this to ft/s2.

[3] The conversion between Canadian and US dollars is currently C$1.55=US$1.00. Gasoline in Canada costs C$0.80 per liter; what is the cost per gallon in US$? (1 liter is 0.264 gallons.)

[4] The light-year is the distance that light travels (in vacuum) in one year. The speed of light is 3.00 × 108m/s. What is one light-year in meters?

[5] The standard class hour is 50 minutes. What is this in microcenturies?

[1] 2.54 cm = 1 in. What is 1 meter in inches? 1 m 100 cm
1 m
1 in
2.54cm
= 100
2.54cm = 39.37 in
[2] g = 9.80 m/s2_{. Convert this to ft/s}2_{.} _{9.8}m
s2
100 cm
1 m
1 in
2.54cm
1 ft
12 in
= 32.2ft
s2

[3] The conversion between Canadian and US dollars is currently C$1.55=US$1.00. Gasoline in Canada costs C$0.60 per liter; what is the cost per gallon in US$? (1 liter is 0.264 gallons.)

0.80 C$ liter 1 liter 0.264 gallons 1 US$ C$1.55 = US$1.95 per gallon

[4] The light-year is the distance that light travels (in vacuum) in one year. The speed of light is 3.00 × 108m/s. What is one light-year in meters?

3.00 × 108m
s · 1 yr
365.24 days
1 yr
·
24 hours
1 day
3600 s
1 hr
= 9.47 × 1015_{m}

[5] The standard class hour is 50 minutes. What is
this in microcenturies?
50 min
1 hr
60 min
1 day
24hr
1 yr
365.24 days
·
1 century
100 yr
106_{microcenturies}
1 century
= 0.95 microcenturies

Practice Set 2

[1] What is 60 mi/hr in ft/sec?

[2] Given 2.54 cm = 1 in exactly, what is 1 mile in kilo-meters?

[3] Water flows down a river at a rate of 500 cubic me-ters per second. What is this in gallons per hour? (1 liter = 10−3m3, 1 liter = 0.264 gallons.)

[4] g = 9.8 m/s2 is the standard gravitational field at the surface of the Earth.

Convert this to light-years/year2.

[5] Astronomers most often use the unit called a “par-sec,” which is about 3.26 light-years.

The farthest objects ever observed are measured to be about 1010light-years away. What is this in mega-parsecs?

[1] The density of water is 1 gm/cm3. Convert this to kg/m3?

[2] Start with 10 m3/2 and put the meters into centime-ters.

[3] If you go on a diet, and lose at a rate of 1 kg/week, what is this in milligrams per second?

[4] The speed of light is 3.00 × 108m/s. What is it in feet per nanosecond?

[5] A certain physical constant is 1.57 gm1/2cm−3/2. Convert the grams to kilograms and the centimeters to meters.