Most students in physical science seem to have more difficulties with problems than with any other part of the course. This is partly because problems involve mathematical analysis, but it is also because there is a certain systematic approach to understanding science that is different than the approach used in most other areas of study. Data must be analyzed in a step-by-step fashion that has been the backbone of science for many years and has come to be known as the scientific method.

In the first few chapters, we will try to lead you through some examples that show how to set up a problem from the information given and how to calculate an answer that is not only numerically correct but also has the correct units and the proper number of significant figures. Problem solving is not always easy, and you may make some mistakes at the beginning. This is all part of the learning process. Hopefully, by the time you get to your first examination, you will be reasonably comfortable with problem solving and the systematic analysis of data.

Let's begin with some examples showing how to use conversion factors. You will find substantial help with conversion factors in Section 1.5 of your textbook.

1. How many meters are there in three kilometers?

The conversion factor for meters to kilometers is:

1 kilometer = 1000 meters

First, we must make the conversion factor into a fraction that is equal to one (1) in such a way that the units in the known quantity (in this case kilometers) will cancel out when we multiply by the fraction. Any quantity multiplied by one (1) will still be numerically the same, so multiplying by the equation that we made equal to one (1) will only change the units, as we desire it to. Here the "kilometer" in the conversion factor must be in the denominator to cancel the unit "kilometer" in the numerator of the quantity we wish to convert. The correct conversion factor is:

1000 meters	= 1
1 kilometer

Applying this to our known length measurement, 3 kilometers, we obtain:

3 kilometers x		1000 meters		= 3000 meters
		1 kilometer

Do you see how the units "kilometer" cancel and only meters are left? This is a trivial example. Most people would just say, since there are 1000 meters in one kilometer, there must be 3000 meters in three kilometers. But what happens if you need to do a more complicated conversion? The above example serves as a guide. This time, however, we will use a "chain" of several conversion factors to get the final answer.

2. If the speedometer on your car reads 55 miles/hour, how fast are you traveling in feet/second?

The length unit "mile" must be converted into "feet" and the time unit "hour" must be converted into "seconds." We will use the following conversion factors:

1 mile = 5280 feet,

1 hour = 60 minutes,

and 1 minute = 60 seconds

The conversion fractions will be:

5280 feet	= 1		1 hour	= 1		1 minute	= 1
1 mile			60 minutes			60 seconds

These were set up so that the units miles, hours, and minutes will all cancel, and we will be left with only feet/second as required in the problem. Look at the next line of equations carefully to see how the conversion ratios have been set up to cancel out the unwanted units. 55 miles/hour means 55 miles in one (1) hour, so we can write the equation as follows:

55 miles	x	5280 feet	x	1 hour	x	1 minute	=	294400 feet	=	81 feet/second
1 hour		1 mile		60 minutes		60 seconds		3600 seconds

This calculation came out with a value of 80.67 but we should only have two significant figures in the answer, so we have to round it off and our final answer is 81 feet/second.

One more example should give you an even better understanding of the unit conversion process.

3. If the area of a circle is 2.3 square meters, what will be the area of this circle in square inches?

Here again we will use a "chain" of conversion factors.

1 meter = 3.28 feet

and 1 foot = 12 inches

You might have used just one conversion factor to go directly from meters to inches if you could have found one, but this example will show you how to make the conversion in multiple steps as is sometimes necessary.

2.3 meters2	x	3.28 feet	x	3.28 feet	x	12 inches	x	12 inches	=
		1 meter		1 meter		1 foot		1 foot

	=	3563.1821 inches2	=	3600 inches2 (to two significant figures)
		1

Now wait a minute! Why did we have to use the conversion factors twice to make the transformation from SI to British units? In this case, the original unit was "meter²" (square meters or meter x meter), so using the conversion factor once would have only converted one of these meters into feet. The second conversion changes the other meter in the squared term.

This means that any squared units must be converted by using double conversion factors, and cubed units would have to use the conversion factors three times. This is useful to know when tables containing conversion factors for areas or volumes are not readily available.

Significant figures also came into play in this calculation. If we follow the rules for significant figures, the number 2.3 has only two significant figures and so the answer can only have two significant figures. You should carry at least one extra figure through your calculations, realizing that the last figure has some uncertainty in it, and then round back to the proper number of figures in your final answer. Electronic calculators let you carry many figures through your calculations but you MUST NOT leave them all in your answer. No one will argue that 3563.1821 contains more figures than have any real meaning. Even 3563 implies that we know more than we really do about the accuracy of this measurement, so we can use only two significant figures and our final answer becomes 3600 inches², or even better, 3.6 x 10³ inches² if we use proper powers-of-10 notation.

One more question. Since the number 1 in the conversion factor "1 foot = 12 inches" has only one figure, why aren't we restricted to only one significant figure in our answer? In this case the 1 is considered an exact whole number; that is, it cannot be 1.000001 or 0.999999; it must be exactly ONE. This means that we can consider the 1 to be 1.0 or 1.00 or even 1.000, to as many significant figures as we need, and we use only the figures in the "measured" data (2.3 m²) to determine the number of significant figures for our answer. (Someone must have measured or calculated this area at some previous time, so we consider it a "measured" quantity.)

Finally, let's do a problem involving density.

4. Find the density of a liquid if a given sample has a volume of 245 mL and a mass of 166.6 grams.

density = mass/volume

= m/V

= 166.6 g / 245 mL = 0.680 g/mL

If we were to look up that value in a table of densities, we would find that the liquid is probably gasoline. Notice that the zero in this answer is necessary to give three significant figures as required in the calculation. If we left it off, we would not be expressing the answer to three significant figures as it should be. Sometimes, as in this case, a calculator gives you too few significant figures.

Are you ready for something new? We think you are, but before we go on let's go over the material in Chapter 1 with a short-answer review and then try some multiple-choice questions. The answers are given to all of these so that you can check your progress. There are also many good review questions and exercises at the end of the chapter in the textbook that you should look over carefully. We have also included worked-out examples of many of the paired exercises in the textbook to help you if you are having trouble solving them by yourself.

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