Handout for Section 3.1
Conversion Factors
(M. Simoes 2/10/93) (edited by P. Grow 7/15/03)
Any number can be multiplied by 1 and it’s value not be altered. This is the key principle behind
conversion factors, to be able to multiply any number by 1 (or a form of 1) and not change it’s value.
Example 1:
5 1  5
The value of 5 remains.
3
Now substitute 1 
3
3 15
The value of 5 remains.
5 
3 3
4
because 4=4. In this same manner we
4
1 foot
1 inch
can say that 1 is equal to
because 1 foot = 12 inches. Also 1=
, because 1 inch = 2.54
12 inches
2.54 cm
 radians
centimeters, as well as 1=
, because  radians = 180 , etc., etc., etc.
180
We can also treat units as though they were numbers, so we can cancel them out (as though they were
numbers).
We can express the value of 1 in different ways. 1 is equal to
Example 2: Convert 3960 feet to miles.
3960 ft.=
3960 ft
1
We have a fraction with the “ft.” units in the numerator (the top of the fraction). To cancel these units
we need “ft.” units in the denominator, (on the bottom of the fraction). Where can we get this fraction? We
3960 ft
can multiply any number by one (or a form of one) and not change it’s value. Thus we can multiply
by
1
1 mile
3960 ft 1 mile
1 (or
, since 1 mile= 5280 ft.) and not change the value.

 0.75 mile
5280 ft.
1
5280 ft.
Notice that the units of feet cancel (like numbers) and only units of mile are left. This method is easier
and more efficient. Let’s go back to the second case: miles per hour.
Example 3: Let’s say an airplane is flying at 100 mph (100 miles per hour or
100 miles
). What is the rate in
hr.
 ft. 
feet per second 
?
 sec. 
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No part of this work may be reproduced without the prior written consent of the Cerritos College Math Learning Center.
100 miles 5280 ft.
1 hr.
ft.


 146.7
hr.
1 mile 3600 sec.
sec.
 5280 ft. 
Notice that the first conversion factor 
 was written with the mile units on the bottom
 1 mile 
and the feet units on the top because we had to cancel the “miles” of “miles per hour” to be left with
 1 hr. 
units of feet on top. The second conversion factor 
 was written with the hour on top
 3600 sec. 
because we had to cancel the “hour” unit of “100 miles per hour” to be left with seconds on the bottom.
We can use this method to convert any unit to any other unit given the relationship between the units.
Example 4: Knowing that the earth rotates once every 24 hours, how many radians per second does the
earth rotate?
1 rotation 2 radians
1 hr.
rad.


 7.27 105
24 hours 1 rotation 3600 sec.
sec.
Example 5: If we are traveling at 65 miles per hour, how many kilometers per day is this?
65 miles 5280 ft. 12 in. 2.54 cm. 1 meter
1 km.
24 hr.
km.






 2511
1 hour 1 mile 1 ft.
1 in. 100 cm. 1000 meters 1 day
day
As we can see, we can change any set of units to another as long as we know the relationship
between them and as long as we are willing to write out the necessary conversion factors.
Relationships:
1 ft.=12 in.
 radians=180
1 in.=2.54 cm.
360  2 rad.=1 rev.
100 cm.=1 meter
60 sec=1 min.
1000 meters.=1 km.
60 min.=1 hr.
5280 ft.=1 statute mile
24 hr.=1 day
6080 ft.=1 nautical mile
365 days=1 yr.
1 nautical mile
 1 knot
hr.
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No part of this work may be reproduced without the prior written consent of the Cerritos College Math Learning Center.
Convert
To
Answers:
1.
10 mph
2.
12 ft.
3.
8.
4 miles
1
6 yrs.
2
in.
15
sec.
rev.
300
hr.
rev.
5
hr.
11 km.
9.
7 knots
4.
5.
6.
7.
cm.
min.
11. 3 decades
10.
1000
in.
sec.
cm.
1.
kilometers
2.
in.
sec.
365.76 cm.
3.
6.44 km.
4.
3,416,400 minutes
minutes
ft.
min.
rad.
sec.
degrees
min.
ft.
ft.
min.
knots
days
5.
6.
7.
8.
176
ft.
min.
rad.
0.524
sec.
deg.
30
min.
36,089 ft.
75
ft.
min.
10. 0.324 knots
9.
709
11.
10950 days
12.
3706 meters
12.
2 nautical miles meters
13.
57.6 mph
13.
50 knots
mph
14.
21.6 knots
knots
15.
0.133
40 km.
hr.
rad.
15. 50
min.
14.
rev.
sec.
rev.
sec.
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No part of this work may be reproduced without the prior written consent of the Cerritos College Math Learning Center.