2) 0.00418 - three significant figures: the 4, the 1, and the 8. This is a typical type of problem where the student errs by giving five significant figures as the answer.
3) 7.09 x 10¯5 - three significant figures. When a number is written in scientific notation, only significant figures are placed into the numerical portion. If this number were taken out of scientific notation, it would be 0.0000709.
4) 91,600 - three significant figures. The last two zeros are not considered to be significant (at least normally). Suppose you had information that showed the zero in the tens place to be significant. How would you show it to be different from the zero in the ones place, which is not significant? The answer is scientific notation. Here is how it would be written: 9.160 x 104. This CLEARLY indicates the presence of four significant figures.
5) 0.003005- four significant figures. No matter how many zeros there are between two significant figures, all the zeros are to be considered significant. A number like 70.000001 would have 8 significant figures.
6) 3.200 x 109 - four significant figures. Notice the use of scientific notation to indicate that there are two zeros which should be significant. If this number were to be written without scientific notation (3,200,000,000) the significance of those two zeros would be lost and you would - wrongly - say that there were only two significant figures.
11) 3.461728 + 14.91 + 0.980001 + 5.2631
12) 23.1 + 4.77 + 125.39 + 3.581
In each of these two problems, examine the decimal portion only. Find the number with the LEAST number of digits in the decimal portion. In problem 1 it is the 14.91 and in problem 2 it is 23.1.
That means problem 1 will have its answer rounded to the 0.01 place and problem 2 will have its answer rounded to the 0.1 place. The correct answers are 24.61 and 156.8.
13) 22.101 - 0.9307
The answer is 21.170. The first value in the problem, with three significant places to the right of the decimal point, dictates how many significant places to the right of the decimal point in the answer.
14) 0.04216 - 0.0004134
The answer is 0.04175.
15) 564,321 - 264,321
This problem is somewhat artifical. The correct answer is 300,000, BUT all of the significant figures are retained. The most correct way to write the answer would be 3.00000 x 105.
16) (3.4617 x 107) ÷ (5.61 x 10¯4)
The calculator shows 6.1706 x 1010 which then rounds to 6.17 x 1010 - three significant figures. The value which dictates this is in boldface.
17) [(9.714 x 105) (2.1482 x 10¯9)] ÷ [(4.1212) (3.7792 x 10¯5)]
The calculator shows 1.3398 x 101 which then rounds to 13.40 - four significant figures. In this problem pay attention to order of operations, since division is not commutative. There are two ways to do this problem using the calculator: 1) multiply the last two numbers, put the result in memory, multiply the first two, then divide that by what is in memory or 2) multiply the first two numbers then do two divisions.
18) (4.7620 x 10¯15) ÷ [(3.8529 x 1012) (2.813 x 10¯7) (9.50)]
The calculator shows 4.625 x 10¯22, which then rounds to 4.62 x 10¯22 - three significant figures. Notice the use of the rounding with five rule.
19) [(561.0) (34,908) (23.0)] ÷ [(21.888) (75.2) (120.00)]
The calculator shows 2280.3972, which rounds off to 2280, three significant figures. In scientific notation, this answer would be 2.28 x 103.
Note this last use of scientific notation to indicate significant figures where otherwise you might not realize they were significant. For example, 2300 looks like it has only two significant figures, but you know (from the problem) it really has three. How do you show this. One way is to use scientific notation like this: 2.30 x 103. Now the 2.30 portion clearly has three significant figures.