Conversion Factors

If you learned the SI units and prefixes described in Section 1.6 “The International System of Units”, then you know that 1 cm is 1/100th of a meter.

[latex]1\text{ cm}=\frac{1}{100\text{ m}}[/latex] 
or
100 cm = 1 m

Suppose we divide both sides of the equation by 1 m (both the number and the unit):

[latex]\frac{100\text{ cm}}{1\text{ m}}=\frac{1\text{ m}}{1\text{ m}}[/latex] 

As long as we perform the same operation on both sides of the equals sign, the expression remains an equality. Look at the right side of the equation; it now has the same quantity in the numerator (the top) as it has in the denominator (the bottom). Any fraction that has the same quantity in the numerator and the denominator has a value of 1:

We know that 100 cm is 1 m, so we have the same quantity on the top and the bottom of our fraction, although it is expressed in different units. A fraction that has equivalent quantities in the numerator and the denominator but expressed in different units is called a conversion factor.

Here is a simple example. How many centimeters are there in 3.55 m? Perhaps you can determine the answer in your head. If there are 100 cm in every meter, then 3.55 m equals 355 cm. To solve the problem more formally with a conversion factor, we first write the quantity we are given, 3.55 m. Then we multiply this quantity by a conversion factor, which is the same as multiplying it by 1. We can write 1 as 100 cm1 m and multiply:

[latex]3.55\text{ m}\times\frac{100\text{ cm}}{1\text{ m}}[/latex] 

The 3.55 m can be thought of as a fraction with a 1 in the denominator. Because m, the abbreviation for meters, occurs in both the numerator and the denominator of our expression, they cancel out:

[latex]\frac{3.55\cancel{\text{ m}}}{1}\times\frac{100\text{ cm}}{1\cancel{\text{ m}}}[/latex] 

The final step is to perform the calculation that remains once the units have been canceled:

[latex]\frac{3.55}{1}\times\frac{100\text{ cm}}{1}=355\text{ cm}[/latex] 

In the final answer, we omit the 1 in the denominator. Thus, by a more formal procedure, we find that 3.55 m equals 355 cm. A generalized description of this process is as follows:

quantity (in old units) × conversion factor = quantity (in new units)

You may be wondering why we use a seemingly complicated procedure for a straightforward conversion. In later studies, the conversion problems you will encounter will not always be so simple. If you can master the technique of applying conversion factors, you will be able to solve a large variety of problems.

In the previous example, we used the fraction $$\frac{100\text{ cm}}{1\text{ m}}$$ as a conversion factor. Does the conversion factor  $$\frac{1\text{ m}}{100\text{ cm}}$$ also equal 1? Yes, it does; it has the same quantity in the numerator as in the denominator (except that they are expressed in different units). Why did we not use that conversion factor? If we had used the second conversion factor, the original unit would not have canceled, and the result would have been meaningless. Here is what we would have gotten:

[latex]3.55\text{ m} \times \frac{1\text{ m}}{100\text{ cm}}=0.0355  \frac{\text{m}^2}{\text{cm}}[/latex]

For the answer to be meaningful, we have to construct the conversion factor in a form that causes the original unit to cancel out. Figure 1.10 “A Concept Map for Conversions” shows a concept map for constructing a proper conversion. The steps for doing a unit conversion problem are summarized in the margin.

Figure 1.10 A Concept Map for Conversions.This is how you construct a conversion factor to convert from one unit to another.

Significant Figures in Conversions

How do conversion factors affect the determination of significant figures? Numbers in conversion factors based on prefix changes, such as kilograms to grams, are not considered in the determination of significant figures in a calculation because the numbers in such conversion factors are exact. Exact numbersA number that is defined or counted. are defined or counted numbers, not measured numbers, and can be considered as having an infinite number of significant figures. (In other words, 1 kg is exactly 1,000 g, by the definition of kilo-.) Counted numbers are also exact. If there are 16 students in a classroom, the number 16 is exact. In contrast, conversion factors that come from measurements (such as density, as we will see shortly) or are approximations have a limited number of significant figures and should be considered in determining the significant figures of the final answer.

13.6 g mercury = 1 mL mercury

This relationship can be used to construct two conversion factors:

[latex]\frac{13.6\text{ g}}{1\text{ mL}}\text{ and }\frac{1\text{ mL}}{13.6\text{ g}}[/latex]

Which one do we use? It depends, as usual, on the units we need to cancel and introduce. For example, suppose we want to know the mass of 16 mL of mercury. We would use the conversion factor that has milliliters on the bottom (so that the milliliter unit cancels) and grams on top so that our final answer has a unit of mass:

[latex]16\cancel{\text{ mL}}\times \frac{13.6\text{ g}}{1\text{ mL}}=217.6\text{ g}=220\text{ g}[/latex]

In the last step, we limit our final answer to two significant figures because the volume quantity has only two significant figures; the 1 in the volume unit is considered an exact number, so it does not affect the number of significant figures. The other conversion factor would be useful if we were given a mass and asked to find volume, as the following example illustrates.

To do a stepwise conversion, we first convert the given amount to the base unit. In this example, the base unit is meters. We know that there are 1,000 m in 1 km:

[latex]54.7\cancel{\text{ km}}\times \frac{1,000\text{ m}}{1\cancel{\text{ km}}}=54,700\text{ m}[/latex]

Then we take the result (54,700 m) and convert it to millimeters, remembering that there are 1,000 mm for every 1 m:

[latex]54,700\cancel{\text{ m}}\times \frac{1,000\text{ m}}{1\cancel{\text{ m}}}=54,700,000\text{ mm}=5.47\times10^7\text{ mm}[/latex]

We have expressed the final answer in scientific notation.

As a shortcut, both steps in the conversion can be combined into a single, multistep expression:

[latex]54.7\cancel{\text{ km}}\times \frac{1,000\cancel{\text{ m}}}{1\cancel{\text{ km}}}\times\frac{1,000\text{ mm}}{1\cancel{\text{ m}}}\times=54,700,000\text{ mm}=5.47\times10^7\text{ mm}[/latex]

Either method—one step at a time or all the steps together—is acceptable. If you do all the steps together, the restriction for the proper number of significant figures should be done after the last step. As long as the math is performed correctly, you should get the same answer no matter which method you use.