Factor Labels

The Factor-Label Method is a very important concept in science. While it may "feel" strange at first, once you gain experience, you will agree that it makes calculations much simpler! In this technique, all measurements and conversion factors are represented as fractions. The investigator, must decide a starting point and an ending point of the calculations. They then add factors, cancelling units, until the starting unit is been converted into the ending unit. Once the factors are arranged correctly, all numerators are multiplied and denominators are divided.

For example, the density of a solid may have a value of 1.35 g/mL. This could be written as:

The first case would indicate multiplication of the density with "1.35 g" as the numerator and "mL" as the denominator. Although, if we decided to divide by the density, the second inverted case would be used with '1 mL' being the new numerator and '1.35 g' being the new denominator.

Therefore, in the Factor-Label method each factor is analyzed one step at a time so that the overall units, after cancelation, are converted from the starting measurement to the ending result. Once the factors are all in place, the numerators are then multiplied and the denominators are divided.

For example, optical fibers 0.10 mm thick can be inserted into a patient’s vein and maneuvered to the heart, where they deliver laser light that breaks up blood clots.  What is the thickness of these optical fibers in inches?

i) the conversion is mm to inches.

Write the starting measurement (0.10 mm), convert "mm to cm" and finally convert "cm into in". Ignoring any multiplication or division by '1', the final result would be found by entering '0.10' in the calculator and dividing by '10' and then dividing by '2.54'. The final result '0.0039' must be accompanied by units or it is meaningless!

A second harder example is as follows (don't worry if you don't understand the chemistry, concentrate on the mathematics), what is the new concentration of HCl if 35.0 mL of 0.250 mol HCl/L solution are diluted to 0.500 L?

Starting value is '35.0 mL' of solution, convert 'mL to L', covert 'L to moles HCl' and then DIVIDE by final volume '0.500 L'. Notice, when you need to divide, place the value to be divided in the denominator.

Using Percentages in Factor Label

Percentages should always be represented as parts per hundred in Factor Label problems. For example if a compound is 25.0 % (m/m) sodium. (m = mass, v = volume) the factor used should be either

Here is an example, An ore of tin contains 35.67 % (m/m) tin. If you need to obtain 45.3 g of tin, how many kilograms of ore must be refined? Assume 100 % recovery of the tin.

First write the percentage as a factor.

Next decide the starting and ending units of the problem. The starting value is almost always something you can touch or pick up. In this case it is 45.3 g of tin. The ending needs to be in "kg ore" (defined by problem!).

The mathematics would be 45.3 multiplied by 100, divided by 35.67, divided by 1000. Notice, '1''s are ignored for they are the identity element and don't change the overall values!

For a second example,

How many kilograms of Tin are present in 100.0 kg of tin ore (the ore contains 35.67 % (m/m) tin).

Note, the % (m/m) indicates ANY unit mass. In the above example, grams were used. In this example, kg are used. It does not matter as long as you are consistent.