1. To start the tutorial, push the f5 key in the upper row on your keyboard f5 Then you can use the up and down arrow keys to advance backwards and forwards through the tutorial.

2. Calculations with significant figures So now you know how to determine the number of significant figures in a number and how to round numbers off, what good does that do?

3. Calculations with significant figures Well, let’s consider the rectangle below. If we want to determine the area of the rectangle, the easiest way would be to measure the length and the height, and then multiply these two numbers together. (For rectangles, Area = Length x Height.)

4. Calculations with significant figures First let’s measure the length: a correct reading might be something like 36.3 cm (…or 36.2 cm or 36.4 cm) 010 cm 20 30 40 50

5. Calculations with significant figures Now let’s measure the height: for height, a correct reading might be some- thing like 6.7 cm (…or 6.6 cm or 6.8 cm). 010 cm 20

6. Calculations with significant figures So let’s multiply these two numbers:

7. Calculations with significant figures So let’s multiply these two numbers: 36.3 x 6.7 = 243.21 (and cm x cm = cm2). So we have 243.21 cm 2. But if we state the area to be 243.21cm 2, we are stating a pretty high level of precision.

8. Calculations with significant figures But if we say the area is 243.21cm 2, we are saying we know the area to a very high level of precision.

9. Calculations with significant figures But if we say the area is 243.21cm 2, we are saying we know the area to a very high level of precision. We are saying that we are certain of the “243.2..” and that we are guessing the “1.”

10. Calculations with significant figures But if 36.2 cm long and 6.6 cm high were also correct measurements, then 36.2 cm x 6.6 cm = 238.92 cm 2 would have to be a correct area for the same rectangle.

11. Calculations with significant figures But if 36.2 cm long and 6.6 cm high were also correct measurements, then 36.2 cm x 6.6 cm = 238.92 cm 2 would have to be a correct area for the same rectangle. That implies we are certain of the “238.9…” and only guessing the “2.”

12. Calculations with significant figures And if 36.4 cm long and 6.8 cm high were also correct measurements, then 36.4 cm x 6.8 cm = 247.52 cm 2 would also have to be a correct area for the same rectangle.

13. Calculations with significant figures And if 36.4 cm long and 6.8 cm high were also correct measurements, then 36.4 cm x 6.8 cm = 247.52 cm 2 would also have to be a correct area for the same rectangle. That implies we are certain of the “247.5…” and only guessing the “2.”

14. Calculations with significant figures 243.21 cm 2, 238.92 cm 2 and 247.52 cm 2

15. Calculations with significant figures 243.21 cm 2, 238.92 cm 2 and 247.52 cm 2 These three values cannot all be correct.

16. Calculations with significant figures 243.21 cm 2, 238.92 cm 2 and 247.52 cm 2 These three values cannot all be correct. The only digit that seems to be definite is the first “2” (in the hundreds place). After that the values are not at all consistent with one another.

17. Calculations with significant figures This would mean that our guess should be the second digit (in the tens place), and that the values should all be rounded there – to two significant figures.

18. Calculations with significant figures 243.21 cm 2 rounds to 240 cm 2 238.92 cm 2 rounds to 240 cm 2 and 247.52 cm 2 rounds to 250 cm 2.

19. Calculations with significant figures 240 cm 2, 240 cm 2 and 250 cm 2.

20. Calculations with significant figures 240 cm 2, 240 cm 2 and 250 cm 2. These are all consistent with one another.

21. Calculations with significant figures 240 cm 2, 240 cm 2 and 250 cm 2. These are all consistent with one another. They all have two significant figures, and they show disagreement only in the guessed digit.

22. Calculations with significant figures Is there a way we could have known from the beginning that our answer needed to be rounded to only two significant figures?

23. Calculations with significant figures If we look at the original measurements that went into the calculation, we see a length of 36.3 cm, which has three significant figures, and a height of 6.7 cm, which has two significant figures.

24. Calculations with significant figures Imagine there is a chain that is made of only two links, and one link is able to hold 3 kg before it breaks and the other is able to hold 2 kg, how much weight can the entire chain hold? Strong enough to hold 3 kg Strong enough to hold 2 kg

25. Calculations with significant figures If you are thinking that the chain could hold 5 kg (3 kg + 2 kg), then think again! Strong enough to hold 3 kg Strong enough to hold 2 kg

26. Calculations with significant figures If you are thinking that the chain could hold 5 kg (3 kg + 2 kg), then think again! The chain would break at its weakest point. And so, as a whole, the chain would only be able to hold 2 kg before it broke. Together only strong enough to hold 2 kg

27. Calculations with significant figures There is an old expression that says: “A chain is only as strong as its weakest link.”

28. Calculations with significant figures There is an old expression that says: “A chain is only as strong as its weakest link.” If a chain were made of ten links, and nine of those links could hold 100 kg, but one could only hold 1 kg…

29. Calculations with significant figures How much weight would the entire chain be able to hold?

30. Calculations with significant figures How much weight would the entire chain be able to hold? Just 1 kg!

31. Calculations with significant figures How much weight would the entire chain be able to hold? Just 1 kg! Essentially, the one weak link ruins it for the rest of the links.

32. Calculations with significant figures The same holds true for calculations involving measurements. Consider the calculation below. 23.40 cm x 0.47 cm x 6.05 cm = precise to 4 significant figures precise to 2 significant figures precise to 3 significant figures

33. Calculations with significant figures The calculator answer has 6 significant figures. 23.40 cm x 0.47 cm x 6.05 cm = 66.5379 cm 3 precise to 4 significant figures precise to 2 significant figures precise to 3 significant figures

34. Calculations with significant figures The calculator answer has 6 significant figures. But the weakest measurement has only 2 significant figures. 23.40 cm x 0.47 cm x 6.05 cm = 66.5379 cm 3 precise to 4 significant figures precise to 2 significant figures precise to 3 significant figures

35. Calculations with significant figures The calculator answer has 6 significant figures. But the weakest measurement has only 2 significant figures. This means the answer must be rounded to only two significant figures: 23.40 cm x 0.47 cm x 6.05 cm = 66.5379 cm 3 precise to 4 significant figures precise to 2 significant figures precise to 3 significant figures 67 cm 3

36. Calculations with significant figures When you report an answer to be something like “66.5379 cm 3 ”(just because that is what showed up on your calculator), you are claiming a level of precision much higher than the measurements deserve. 23.40 cm x 0.47 cm x 6.05 cm = 66.5379 cm 3 precise to 4 significant figures precise to 2 significant figures precise to 3 significant figures 67 cm 3

37. Calculations with significant figures An answer of 66.5379 cm 3 means that the “66.537..” are definite, and only the “9” is a guess. 23.40 cm x 0.47 cm x 6.05 cm = 66.5379 cm 3 precise to 4 significant figures precise to 2 significant figures precise to 3 significant figures 67 cm 3

38. Calculations with significant figures An answer of 66.5379 cm 3 means that the “66.537..” are definite, and only the “9” is a guess. But if the 0.47 cm could have just as easily been read as 0.46 cm, consider how different the answer would be. 23.40 cm x 0.47 cm x 6.05 cm = 66.5379 cm 3 precise to 4 significant figures precise to 2 significant figures precise to 3 significant figures 67 cm 3

39. Calculations with significant figures So here is the rule: When multiplying or dividing two or more measurements, always round your answer off to the number of significant figures in the weakest measurement. (The weakest measurement is the one with the fewest significant figures)

40. Calculations with significant figures So here is the rule: When multiplying or dividing two or more measurements, always round your answer off to the number of significant figures in the weakest measurement. (The weakest measurement is the one with the fewest significant figures) This ensures that your answer will not be any more or less precise than it should be.

41. Calculations with significant figures If crude measurements were made, then only crude values can be calculated from them. If more precise measurements were made, then more precise values can be calculated.

42. Calculations with significant figures So let’s say a student is calculating the average speed of a car as it traveled down the road.

43. Calculations with significant figures So let’s say a student is calculating the average speed of a car as it traveled down the road. Speed is distance divided by time.

44. Calculations with significant figures So let’s say a student is calculating the average speed of a car as it traveled down the road. Speed is distance divided by time. The student measures the time with a very precise stop watch and records a time of 146.39 s.

45. Calculations with significant figures So let’s say a student is calculating the average speed of a car as it traveled down the road. Speed is distance divided by time. The student measures the time with a very precise stop watch and records a time of 146.39 s. Distance is measured rather crudely: 680 m. 0100200300400500600700800 m

46. Calculations with significant figures Speed = = = 0100200300400500600700800 m distance time 680 m 146.39 s

47. Calculations with significant figures Speed = = = 4.645126033 m/s 0100200300400500600700800 m distance time 680 m 146.39 s

48. Calculations with significant figures Speed = = = 4.645126033 m/s This answer is what appears on the calculator, but it is obviously way too precise. 0100200300400500600700800 m distance time 680 m 146.39 s

49. Calculations with significant figures Speed = = = 4.645126033 m/s This answer is what appears on the calculator, but it is obviously way too precise. What should it be rounded to? 0100200300400500600700800 m distance time 680 m 146.39 s

50. Calculations with significant figures Speed = = = 4.645126033 m/s The distance (680 m) has two significant figures 0100200300400500600700800 m distance time 680 m 146.39 s

51. Calculations with significant figures Speed = = = 4.645126033 m/s The distance (680 m) has two significant figures… and the time (146.39 s) has five significant figures. 0100200300400500600700800 m distance time 680 m 146.39 s

52. Calculations with significant figures Speed = = = 4.645126033 m/s The distance (680 m) has two significant figures… and the time (146.39 s) has five significant figures. The weaker measurement is the one with just two significant figures... 0100200300400500600700800 m distance time 680 m 146.39 s

53. Calculations with significant figures Speed = = = 4.645126033 m/s The distance (680 m) has two significant figures… and the time (146.39 s) has five significant figures. The weaker measurement is the one with just two significant figures… so the answer should be rounded to just two sig. figs. 0100200300400500600700800 m distance time 680 m 146.39 s

54. Calculations with significant figures Speed = = = 4.645126033 m/s The distance (680 m) has two significant figures… and the time (146.39 s) has five significant figures. The weaker measurement is the one with just two significant figures… so the answer should be rounded to just two sig. figs. 0100200300400500600700800 m distance time 680 m 146.39 s 4.6 m/s

55. Calculations with significant figures Now try each of the following 20 problems.

56. Calculations with significant figures Now try each of the following 20 problems. Use a calculator, and then write your answer on paper.

57. Calculations with significant figures Now try each of the following 20 problems. Use a calculator, and then write your answer on paper. Make sure to round the answer to the correct number of significant figures…

58. Calculations with significant figures Now try each of the following 20 problems. Use a calculator, and then write your answer on paper. Make sure to round the answer to the correct number of significant figures… …and also make sure to include correct units with each answer.

59. Calculations with significant figures 1) 34.6 m x 14.8 m =

60. Calculations with significant figures 1) 34.6 m x 14.8 m = 512 m 2

61. Calculations with significant figures 1) 34.6 m x 14.8 m = 512 m 2 The 34.6 m and the 14.8 m both have three significant figures, so the answer is rounded to three significant figures.

62. Calculations with significant figures 2) 67 cm x 38 cm =

63. Calculations with significant figures 2) 67 cm x 38 cm = 2500 cm 2

64. Calculations with significant figures 2) 67 cm x 38 cm = 2500 cm 2 The 67 cm and the 38 cm both have two significant figures, so the answer must be rounded to two significant figures.

65. Calculations with significant figures 3) 0.0057 m x 0.0063 m =

66. Calculations with significant figures 3) 0.0057 m x 0.0063 m = 0.000036 m 2

67. Calculations with significant figures 3) 0.0057 m x 0.0063 m = 0.000036 m 2 The 0.0057 m and the 0.0063 m both have two significant figures, so the answer must be rounded to two significant figures.

68. Calculations with significant figures 4) 34.0 mm x 72.7 mm =

69. Calculations with significant figures 4) 34.0 mm x 72.7 mm = 2470 mm 2

70. Calculations with significant figures 4) 34.0 mm x 72.7 mm = 2470 mm 2 The 34.0 mm and the 72.7 mm both have three significant figures, so the answer is rounded to three significant figures.

71. Calculations with significant figures 5) 18.38 m x 23.54 m =

72. Calculations with significant figures 5) 18.38 m x 23.54 m = 432.7 m 2

73. Calculations with significant figures 5) 18.38 m x 23.54 m = 432.7 m 2 The 18.38 m and the 23.54 m both have four significant figures, so the answer is rounded to four significant figures.

74. Calculations with significant figures 6) 207 m x 64 m =

75. Calculations with significant figures 6) 207 m x 64 m = 13,000 m 2

76. Calculations with significant figures 6) 207 m x 64 m = 13,000 m 2 The 207 m has three significant figures, but the 64 m has only two. Thus, the answer should be rounded to two significant figures.

77. Calculations with significant figures 7) 296.3 cm x 0.8 cm =

78. Calculations with significant figures 7) 296.3 cm x 0.8 cm = 200 cm 2

79. Calculations with significant figures 7) 296.3 cm x 0.8 cm = 200 cm 2 The 296.3 cm has four significant figures, but the 0.8 cm only has one. Thus, the answer must be rounded to just one significant figure.

80. Calculations with significant figures 8) 3.4 m x 16.3 m x 25.7 m =

81. Calculations with significant figures 8) 3.4 m x 16.3 m x 25.7 m = 1400 m 3

82. Calculations with significant figures 8) 3.4 m x 16.3 m x 25.7 m = 1400 m 3 The 3.4 m has two significant figures, and the 16.3 m and 25.7 m each have three significant figures. Thus, the answer is rounded to just two significant figure.

83. Calculations with significant figures 9) 0.48 mL x 9.365 g/mL =

84. Calculations with significant figures 9) 0.48 mL x 9.365 g/mL = 4.5 g

85. Calculations with significant figures 9) 0.48 mL x 9.365 g/mL = 4.5 g The 0.48 mL has two significant figures, and the 9.365 cm has four. Thus, the answer is rounded to just two significant figures.

86. Calculations with significant figures 10) 25.50 m x 12.00 m =

87. Calculations with significant figures 10) 25.50 m x 12.00 m = 306.0 m 2

88. Calculations with significant figures 10) 25.50 m x 12.00 m = 306.0 m 2 The 25.50 m and the 12.00 m both have four significant figures, so the answer should have four significant figures. The calculator gives only “306” as the answer. In this situation, the answer must be enhanced up to four significant figures: “306.0”

89. Calculations with significant figures 11) 17.5 km - 3.20 hr =

90. Calculations with significant figures 11) 17.5 km - 3.20 hr = 5.47 km/hr

91. Calculations with significant figures 11) 17.5 km - 3.20 hr = 5.47 km/hr The 17.5 km and the 3.20 hr both have three significant figures, so the answer should have just three significant figures.

92. Calculations with significant figures 12) 234.53 g - 85 mL =

93. Calculations with significant figures 12) 234.53 g - 85 mL = 2.8 g/mL

94. Calculations with significant figures 12) 234.53 g - 85 mL = 2.8 g/mL The 234.53 g has five significant figures, but the 85 mL has only two. Thus, the answer should have only two significant figures.

95. Calculations with significant figures 13) 2300 m 2 - 725 m =

96. Calculations with significant figures 13) 2300 m 2 - 725 m = 3.2 m

97. Calculations with significant figures 13) 2300 m 2 - 725 m = 3.2 m The 2300 m 2 has two significant figures and the 725 m has three. The answer should therefore be rounded to just two significant figures.

98. Calculations with significant figures 14) 17.5 km - 3.50 hr =

99. Calculations with significant figures 14) 17.5 km - 3.50 hr = 5.00 km/hr

100. Calculations with significant figures 14) 17.5 km - 3.50 hr = 5.00 km/hr The 17.5 km and the 3.50 hr both have three significant figures, so the answer should have three significant figures. The calculator gives an answer of simply “5.” So this “5” must be enhanced up to three significant figures: “5.00.”

101. Calculations with significant figures 15) 32.0 m 3 – 7.0 m =

102. Calculations with significant figures 15) 32.0 m 3 – 7.0 m = 4.6 m 2

103. Calculations with significant figures 15) 32.0 m 3 – 7.0 m = 4.6 m 2 The 32.0 m 3 has three significant figures, but the 7.0 m has only two, so the answer should be rounded to just two significant figures.

104. Calculations with significant figures 16) 679.0 g – 678.9 mL =

105. Calculations with significant figures 16) 679.0 g – 678.9 mL = 1.000 g/mL

106. Calculations with significant figures 16) 679.0 g – 678.9 mL = 1.000 g/mL The 679.0 g and the 678.9 mL both have four significant figures, so the answer should be rounded to four significant figures.

107. Calculations with significant figures 17) (65 m x 17 m) – 4.83 s =

108. Calculations with significant figures 17) (65 m x 17 m) – 4.83 s = 230 m 2 /s

109. Calculations with significant figures 17) (65 m x 17 m) – 4.83 s = 230 m 2 /s The 65 m and 17 m both have two significant figures, and the 4.83 s has three significant figures. Thus the answer should have two significant figures.

110. Calculations with significant figures 18) 56.94 g – (3.42 cm x 7.61 cm x 0.35 cm) =

111. Calculations with significant figures 18) 56.94 g – (3.42 cm x 7.61 cm x 0.35 cm) = 6.3 g/cm 3

112. Calculations with significant figures 18) 56.94 g – (3.42 cm x 7.61 cm x 0.35 cm) = 6.3 g/cm 3 The 56.94 g has four significant figures, the 3.42 cm and the 7.61 cm each have three significant figures, but the 0.35 cm has only two significant figures. The answer therefore should be rounded to just two significant figures.

113. Calculations with significant figures 19) 215 cm x 372 cm =

114. Calculations with significant figures 19) 215 cm x 372 cm = 80,000 cm 2

115. Calculations with significant figures 19) 215 cm x 372 cm = 80,000 cm 2 The 215 cm and the 372 cm both have three significant figures, so the answer should be rounded to three significant figures. 79,980 rounds up to 80,000, which appears to have only one significant figure. A line over the second 0 fixes this problem.

116. Calculations with significant figures 20) 2.480 m – 0.124 s =

117. Calculations with significant figures 20) 2.480 m – 0.124 s = 20.0 m/s

118. Calculations with significant figures 20) 2.480 m – 0.124 s = 20.0 m/s The 2.480 m has four significant figures and the 0.124 s has three, so the answer should have three significant figures. The calculator gives an answer of “20”, which has only one significant figure. It must be enhanced to three significant figures: 20.0.

119. Calculations with significant figures So… How many did you get correct?

120. Calculations with significant figures So… How many did you get correct? Hopefully this tutorial program has helped you understand how to round off answers for these sort of calculations

121. Calculations with significant figures It is important to remember that this rule of rounding answers off to the fewest number of significant figures applies only to multiplication and division.

122. Calculations with significant figures It is important to remember that this rule of rounding answers off to the fewest number of significant figures applies only to multiplication and division. There is a different rule that is used for adding and subtracting...

123. Calculations with significant figures It is important to remember that this rule of rounding answers off to the fewest number of significant figures applies only to multiplication and division. There is a different rule that is used for adding and subtracting… and that will be the topic of our next tutorial.