Ralph & Bev Shock  

Chapter 3

The Factor-Label Method

The factor-label method is an efficient and easy method of solving a many stepped problem in chemistry or in any other problem area requiring simple mathematics. We will use this method throughout the whole year to work problems.

Consider the following problem:

1.5 miles = ______________ inches

To solve this or any other problem by the factor-label method you move in steps towards the right answer. You might not know how many inches 1.5 miles is, but you should know how many feet to the mile and how many inches to the foot.

Rule 1

Start with your "given". In the factor-label method all the factors are in the form of fractions. To make 1.5 miles into a fraction, put it over "1".

Many times your given is already in the form of a fraction. In that case, just write it down as is. (Example: 1.5 miles/hour = ________________ inches/second). By the way, remember that the word "per" actually means "divided by".

Rule 2

The label of the numerator in the first factor becomes the label in the denominator of the next factor. In other words, bring down the label to the next factor. Never bring down a number to the next factor.

CORRECT INCORRECT

Rule 3

Supply a factor using the label you just brought down as a clue. In this case ask yourself, "What is a mile equal to?" Most students would say, "5280 feet". A few might say, "1760 yards". Either of these factors work equally well. There is always more than one way to do a factor-label problem.

The reason for bringing down the label to the second factor should be clear to you at this time. Since they are on opposite sides of the dividing line, they cancel. Repeat rule 3, factoring one step at a time, until you have factored the problem to the "asked for" units.

Remember! In each factor the numerator must be equivalent to the denominator. 5280 feet is equivalent to 1 mile. 12 inches is equivalent to 1 foot. Never break this rule. Always double check your factors to make sure the numerator is equivalent to the denominator.

Rule 4

To get the correct answer, multiply any numerators and divide by any denominators. Don’t forget to round your answer to the proper number of significant digits. In our example the right answer is determined by doing the following mathematical steps:

1.5 X 5280 X 12 = 95040

1.5 miles = 95,000 inches

By the way, think about your numbers when your looking for your least significant number in your problems. Many conversions have unlimited sig digs even though they may look like they have as few as one. Example: The conversion 1 mile = 5280 feet means that exactly 1.0000000....etc. miles is exactly 5280.0000000...etc. feet.

Sometimes a problem has two parts two it. I suggested one to you earlier:

You already know how to solve the first half of the problem. The only difference here is that 1.5 miles is now over 1 hour rather than just plain 1.

All we have to do to solve this problem now is to change hours to seconds. We do this the same way as we have been doing, BUT you have to bring up the labels now so that they cancel.

In this case to get the correct answer we do the following mathematical operations on our calculator:

1.5 X 5280 X 12 ÷ 60 ÷ 60 = 26.4

Often times the problem will require that you convert the metric system units to the english system (or visa versa)

14 tons = ______________ grams

Rule 5

Use a "bridge" if you have to go from english to metric (or visa versa).

The bridges are:

meters to feet (length) liters to quarts (volume) kilograms to pounds (mass)

Conversions from metric to english for these standards are:

1 m = 3.281 ft. 1 L = 1.057 qts 1 kg = 2.205 lbs

Note: The english portion of these conversions contains only 4 significant digits.

Rule 6

If you have to convert from one metric unit to another (kg to g, in our example) use the following rule:

The unit with the prefix (the label that has 2 letters in it!) gets a "1". The other unit gets the multiplier for what the prefix means.

14 X 2000 ÷ 2.205 X 1000 = 12,727,272

14 tons = 13,000,000 grams

The logic behind this last rule is very sound if you think about it...kilo means 1000, so 1 kilogram means 1000 grams. (1 kg = 1000 grams) If you accidentally invert a factor like this, your answer is off by a million times.

Lastly...sometimes a problem will ask you to go from a unit of cubic measurement to another unit of cubic measurement (13,000 in³ = ___________ mi³) or perhaps even from a cubic measurement to a volume
(13,000 in³ = ____________ pints). In this case remember that as you factor you will have to cube some of the numbers along the way. Example: Are there 12 in³ to 1 ft³? Certainly not! There are 12 X 12 X 12 in³
(or 12 in)³ to 1 X 1 X 1 ft³ (or 1 ft)³.

Note: In switching from a cubed length to a volume, you will have to use the following conversion somewhere in the problem:

1dm3 = 1 L

(Think of it as a bridge between cubed length and volume.)

Rule 7

When converting from one cubic length to another cubic length you have to cube the cubed factors.

82.8 km³ = ________ µm³

82.8 X 1000 X 1000 X 1000 ÷ 1 X 10^-6 ÷ 1 X 10^-6 ÷ 1 X 10^-6 ÷ = 8.28 X 10²⁸

82.8 km³ = 8.28 X 10²⁸ µm³

13,000 in³ = ____________ pints

13,000 ÷ 12 ÷ 12 ÷ 12 ÷ 3.281 ÷ 3.281 ÷ 3.281 ÷ 0.1 ÷ 0.1 ÷ 0.1 X 1.057 X 2 = 450.2833681

13,000 in³ = 450 pints

A SUMMARY OF THE FACTOR-LABEL RULES

1. Write down your given in the form of a fraction

2. Make sure your labels cancel as you move from factor to factor.

3. The numerator and denominator of each factor are equivalent.

4. Numerators are multiplied, denominators are divided.

5. When going from metric to english (or visa versa) use a bridge.

6. If you must convert from one unit to another within the metric system, the unit with the prefix gets a "1" and the other unit gets the multiplier.

7. Cube the numbers when a factor goes from one cubed length to another cubed length.

Students should know the following conversions and prefixes:

12 in = 1 ft 16 oz = 1 pt 16 oz = 1 lb 60 sec = 1 min
3 ft = 1 yd 2 pt = 1 qt 2000 lbs = 1 ton 60 min = 1 hr
5280 ft = 1 mi qt = 1 gal 20 scruples = 1 grain 24 hr = 1 day
5.5 yds (exactly = 1 rod 8 qt = 1 peck 3 scruples = 1 dram 365.25 days = 1 year
40 rods = 1 furlong 4 pecks = 1 bushel 96 drams = 1 lb
Important Metric to English and English to Metric conversions:

Mass: 2.205 lbs/kg, 453.59 g/lb

Length: 1.609 km/mi, 2.54 cm/in, 3.281 ft/m

Volume: 3.785 L/gal.

Less than one
deci d .1 1 X 10^-1 tenth
centi c .01 1 X 10^-2 hundredth
milli m .001 1 X 10^-3 thousandth
micro µ .000 001 1 X 10^-6 millionth
nano n .000 000 001 1 X 10^-9 billionth
pico p .000 000 000 001 1 X 10^-12 trillionth

Greater than one
deka da 10 1 X 10¹ ten
hecta H 100 1 X 10² hundred
kilo k 1000 1 X 10³ thousand
mega M 1,000,000 1 X 10⁶ million
giga G 1,000,000,000 1 X 10⁹ billion
tera T 1,000,000,000,000 1 X 10¹² trillion

Note: For very large or very small numbers ( > 1 X 10³, < 1 X 10^-3) it is more convenient to use scientific notation on your calculator and for showing your work on paper. For numbers between 1 X 10³ and 1 X 10^-3 it is more convenient to use regular notation (i.e. 1000 to .001) on your calculator and paperwork.