The factor label method
• A way to solve math problems in science
• Used to convert km to miles, m to km, mol to g, g to mol, etc.
• To use this we need: 1) desired quantity, 2) given quantity, 3) conversion factors
• Conversion factors are valid relationships or equities expressed as a fraction
E.g. for 1 km=0.6 miles the conversion factor is
1 km
0.6
miles or
0.6
miles
1 km
Q. write conversion factors for 1 foot =12 inches
Q. what conversion factors can you think of that involve meters?

Conversion factors
Conversion factors for 1 ft = 12 in
1
12 foot inches or
12 inches
1 foot
There are almost an infinite number of conversion factors that include meters:
1000 m
1 km
,
1 m
100 cm
,
1 m
1000 mm
1 m
3.28
feet
,
1 m
39.37
inches
,
0.9144
yards
1 m

The steps to follow
Now we are ready to solve problems using the factor label method. The steps involved are:
1. Write down the desired quantity/units
2. Equate the desired quantity to given quantity
3. Determine what conversion factors you can use (both universal and question specific)
4. Multiply given quantity by the appropriate conversion factors to eliminate units you don’t want and leave units you do want
5. Complete the math

Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km
First write down the desired quantity

Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47 mi
Next, equate desired quantity to the given quantity

Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47 mi
Now we have to choose a conversion factor

Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47 mi
1 km
0.621 mi
0.621 mi
1 km
What conversion factors are possible?

Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47 mi
1 km 0.621 mi
0.621 mi 1 km
Pick the one that will allow you to cancel out miles

Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47 mi
1 km 0.621 mi
0.621 mi 1 km
Pick the one that will allow you to cancel out miles

Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47 mi
1 km
0.621 mi
0.621 mi
1 km
Multiply given quantity by chosen conversion factor

Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47 mi x 1 km
0.621
mi
Multiply given quantity by chosen conversion factor

Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47 mi x 1 km
0.621
mi
Cross out common factors

Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47 x 1 km
0.621
Cross out common factors

Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47 x 1 km
0.621
Are the units now correct?

Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47 x 1 km
0.621
Yes. Both sides have km as units.

Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47 x 1 km
0.621
Yes. Both sides have km as units.

Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47 x 1 km
0.621
= 75.7 km
Now finish the math.

Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47 x 1 km
0.621
= 75.7 km
The final answer is
75.7 km

Summary
The previous problem was not that hard
In other words, you probably could have done it faster using a different method
However, for harder problems the factor label method is easiest

More examples
1. You want to buy 100 U.S. dollars. If the exchange rate is 1 Can$ = 0.65 US$, how much will it cost?
# Can$ = 100 US$ x 1 Can$
0.65 US$
= 153.85 Can$

More examples
2. There are 12 inches in a foot, 0.394 inches in a centimeter, and 3 feet in a yard. How many cm are in one yard?
# cm = 1 yd x 3 ft
1 yd x 12 in
1 ft x 1 cm
0.394 in
= 91.37 cm

Assignment
Answer questions using the factor label method:
1. Calculate how many feet are in 1 meter (use information from the examples above.
2. How many decimeters are in 2 kilometers?
3. Show how many mL are in 1 L?
4. If Nogales is 35.34 miles from Sahuarita, calculate how far that is in km. (1km=0.62mi)
5. Now calculate how many cm Nogales is from Sahuarita.