Multiplication and Division

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In mathematical operations involving significant figures, the answer is reported in such a way that it reflects the reliability of the least precise operation. Let's state that another way: a chain is no stronger than its weakest link. An answer is no more precise that the least precise number used to get the answer. Let's do it one more time: imagine a team race where you and your team must finish together. Who dictates the speed of the team? Of course, the slowest member of the team. Your answer cannot be MORE precise than the least precise measurement.

The following rule applies for multiplication and division:

The LEAST number of significant figures in any number of the problem determines the number of significant figures in the answer.

This means you MUST know how to recognize significant figures in order to use this rule.

Example #1: 2.5 x 3.42.

The answer to this problem would be 8.6 (which was rounded from the calculator reading of 8.55). Why?

2.5 has two significant figures while 3.42 has three. Two significant figures is less precise than three, so the answer has two significant figures.

Example #2: How many significant figures will the answer to 3.10 x 4.520 have?

You may have said two. This is too few. A common error is for the student to look at a number like 3.10 and think it has two significant figures. The zero in the hundedth's place is not recognized as significant when, in fact, it is. 3.10 has three significant figures.

Three is the correct answer. 14.0 has three significant figures. Note that the zero in the tenth's place is considered significant. All trailing zeros in the decimal portion are considered significant.

Another common error is for the student to think that 14 and 14.0 are the same thing. THEY ARE NOT. 14.0 is ten times more precise than 14. The two numbers have the same value, but they convey different meanings about how trustworthy they are.

Four is also an incorrect answer given by some ChemTeam students. It is too many significant figures. One possible reason for this answer lies in the number 4.520. This number has four significant figures while 3.10 has three. Somehow, the student (YOU!) maybe got the idea that it is the GREATEST number of significant figures in the problem that dictates the answer. It is the LEAST.

Sometimes student will answer this with five. Most likely you responded with this answer because it says 14.012 on your calculator. This answer would have been correct in your math class because mathematics does not have the significant figure concept.

Example #3: 2.33 x 6.085 x 2.1. How many significant figures in the answer?

Answer - two.

Which number decides this?

Answer - the 2.1.

Why?

It has the least number of significant figures in the problem. It is, therefore, the least precise measurement.

Example #4: (4.52 x 10¯^{4}) ÷ (3.980 x 10¯^{6}).

How many significant figures in the answer?

Answer - three.

Which number decides this?

Answer - the 4.52 x 10¯^{4}.

Why?

It has the least number of significant figures in the problem. It is, therefore, the least precise measurement. Notice it is the 4.52 portion that plays the role of determining significant figures; the exponential portion plays no role.

Solve these, then click the link for answers and discussion.

1) (3.4617 x 10^{7}) ÷ (5.61 x 10¯^{4})

2) [(9.714 x 10^{5}) (2.1482 x 10¯^{9})] ÷ [(4.1212) (3.7792 x 10¯^{5})]. Watch your order of operations on this problem.

3) (4.7620 x 10¯^{15}) ÷ [(3.8529 x 10^{12}) (2.813 x 10¯^{7}) (9.50)]

4) [(561.0) (34,908) (23.0)] ÷ [(21.888) (75.2) (120.00)]

There might come an occasion in chemistry when you are not exactly sure how many significant figures are called for. Suppose the textbook mentions 100 mL. You look at this and see only one significant figure.

However, an experienced chemist would know that 100 mL can be easily measured to 3 or 4 significant figures. Why then doesn't the textbook (or the professor) write 100.0 (for 4 sig figs) or 1.00 x 10^{2} (for 3 sig figs)?
The textbook writer or the professor might be assuming that all in his or her audience understands these matters and so it is no big deal to simply write 100. Or they are lazy.

So, a brief word of advice. If you haven't a clue as to how many significant figures to use, try using three or four. These are reasonable numbers of significant figures for most chemical activities.

Also, look out for the instructor who ignores significant figures in lecture, then makes a big deal of it on a test. Forewarned is forearmed!

It seems that most high school students treat the numbers on their calculator screen as Holy Writ. NO, IT IS NOT. Most of the string of numbers on the screen does not belong in your final answer.

These two problems are intended to illustrate the difficulty surrounding calculators and significant figures.

1) Calculate the area in cm^{2} of these two samples:

length (cm) width (cm) area (cm^{2}) a. 27.81 20.49 569.8269 b. 27.93 20.36 568.6547

2) Calculate the volume in cm^{3} of the samples from the given thickness:

thickness (cm) volume (cm^{3}) a. 0.710 404.57709 b. 0.690 392.37181

Solution - ROUND OFF!