Conversions
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.
1 ft
(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.
365.24 days
24 hr
60 min
60 sec
1 yr = 1 yr ·
·
·
·
1 yr
1 day
1 hr
1 min
365.24 day
/
24 hr
/
60 min
/
60 sec
= 1 yr
/·
·
·
·
1 yr
/
1 day
/
1 hr
/
1 min
/
= 365.24 · 24 · 60 · 60 · sec = 31556736 sec = 3.1557 × 107 sec,
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 × 107 seconds — easier
to remember.) Another example:
1 mi
1 mi 5280 ft
1 hr
1 min
=
1 hr
1 hr
1 mi
60 min
60 sec
1 mi
/ 5280 ft
1 hr
/
1 min
/
5280 ft
=
=
. = 1.47 ft/sec
1 hr
/
1 mi
/
60 min
/
60 sec
3600 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 ft2
.
12 in
This isn’t wrong , but it’s pretty silly, and isn’t responsive to the original question of converting 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.
1
2
Conversions
To avoid this trap, try writing the units first and fill in the numbers next, as in
ft
yard
5280 ft
1 yard
1 mi ·
·
THEN
1 mi ·
·
= 1760 yards
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 mi
2
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.
1/2
10 cm
·
m 1/2
cm
then
10 cm
1/2
·
1m
100 cm
1/2
10
=√
m1/2 = 1 m1/2 .
100
For a particularly odd unit that I once encountered, convert 10 cm2/3 gm−1/3 sec2 to
meters and kilograms. First, just get the units in place:
−1/3
kg
m 2/3
·
10 cm gm
sec ·
THEN
cm
gm
2/3 −1/3
1 kg
1m
2/3
−1/3
2
·
10 cm gm
sec ·
100 cm
103 gm
10
=
m2/3 kg−1/3 sec2
2/3
100 · 10−1
100
m2/3 kg−1/3 sec2
=
10 · 101/3
10
= 1/3 m2/3 kg−1/3 sec2
10
2/3
−1/3
2
Conversions
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 ×
108 m/s. What is one light-year in meters?
[5] The standard class hour is 50 minutes. What is this
in microcenturies?
3
4
Conversions
Solutions for 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 .
1m
m
9.8 2
s
100 cm
1m
100 cm
1m
1 in
2.54cm
1 in
2.54cm
=
100
cm = 39.37 in
2.54
1 ft
12 in
= 32.2
C$
1 liter
1 US$
[3] The conversion between Canadian and US dollars 0.80
=
liter 0.264 gallons
C$1.55
is currently C$1.55=US$1.00. Gasoline in Canada
US$1.95 per gallon
costs C$0.60 per liter; what is the cost per gallon
in US$? (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 × 108 m/s. What is one light-year in meters?
8m
365.24 days
3.00 × 10
· 1 yr
s
1 yr
24 hours
3600 s
1 day
1 hr
15
= 9.47 × 10 m
[5] The standard class hour is 50 minutes. What is
this in microcenturies?
·
1 hr
1 day
1 yr
50 min
·
60 min
24hr
365.24 days
6
1 century
10 microcenturies
100 yr
1 century
= 0.95 microcenturies
ft
s2
Conversions
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 kilometers?
[3] Water flows down a river at a rate of 500 cubic meters per second. What is this in gallons per hour?
(1 liter = 10−3 m3 , 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 “parsec,” which is about 3.26 light-years.
The farthest objects ever observed are measured to
be about 1010 light-years away. What is this in megaparsecs?
5
6
Conversions
Practice Set 3
[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 centimeters.
[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 × 108 m/s. What is it in
feet per nanosecond?
[5] A certain physical constant is 1.57 gm1/2 cm−3/2 .
Convert the grams to kilograms and the centimeters
to meters.