1. Calculations with significant figures So now you know what to do when you multiply and divide measured quantities:

2. Calculations with significant figures So now you know what to do when you multiply and divide measured quantities: simply round your answer off to the same number of significant figures as there are in the weakest measurement

3. Calculations with significant figures So now you know what to do when you multiply and divide measured quantities: simply round your answer off to the same number of significant figures as there are in the weakest measurement (weakest meaning fewest number of significant figures).

4. Calculations with significant figures So now you know what to do when you multiply and divide measured quantities: simply round your answer off to the same number of significant figures as there are in the weakest measurement (weakest meaning fewest number of significant figures). But do addition and subtraction follow the same rule?

5. Calculations with significant figures So now you know what to do when you multiply and divide measured quantities: simply round your answer off to the same number of significant figures as there are in the weakest measurement (weakest meaning fewest number of significant figures). But do addition and subtraction follow the same rule? Not quite.

6. Calculations with significant figures In adding and subtracting, you still look for the weakest measurement, but weak is defined differently.

7. Calculations with significant figures In adding and subtracting, you still look for the weakest measurement, but weak is defined differently. You dont count significant figures at all.

8. Calculations with significant figures In adding and subtracting, you still look for the weakest measurement, but weak is defined differently. You dont count significant figures at all. Instead, you look at what place the guess (the last significant figure) is in.

9. Calculations with significant figures For example, in 138.4 g, the guess is the 4 and it is in the tenths place.., ten-thousands place thousands place hundreds place tens place hundredths place thousandths place ten-thousandths place 100000000 ones place tenths place

10. Calculations with significant figures For example, in 138.4 g, the guess is the 4 and it is in the tenths place. In 0.281 g, the guess is the 1 and it is in the thousandths place.., ten-thousands place thousands place hundreds place tens place hundredths place thousandths place ten-thousandths place 100000000 ones place tenths place

11. Calculations with significant figures For example, in 138.4 g, the guess is the 4 and it is in the tenths place. In 0.281 g, the guess is the 1 and it is in the thousandths place. Between these two measurements, 138.4 g is the weaker measurement, because its guess is in the higher place.., ten-thousands place thousands place hundreds place tens place hundredths place thousandths place ten-thousandths place 100000000 ones place tenths place

12. Calculations with significant figures And if these two measurements were added together: 138.4 g + 0.281 g

13. Calculations with significant figures And if these two measurements were added together: 138.4 g + 0.281 g 138.681 g This is what the calculator would give as the answer.

14. Calculations with significant figures And if these two measurements were added together: 138.4 g + 0.281 g 138.681 g This is what the calculator would give as the answer. But it should be rounded at the highest guessed place (the tenths place).

15. Calculations with significant figures And if these two measurements were added together: 138.4 g + 0.281 g 138.681 g This is what the calculator would give as the answer. But it should be rounded at the highest guessed place (the tenths place). Which would change it to…

16. Calculations with significant figures And if these two measurements were added together: 138.4 g + 0.281 g 138.7 g This is what the calculator would give as the answer. But it should be rounded at the highest guessed place (the tenths place). Which would change it to…

17. Calculations with significant figures And if these two measurements were added together: 138.4 g + 0.281 g 138.681 g The reason is this: starting at the left-hand side of the calculator answer:

18. Calculations with significant figures And if these two measurements were added together: 138.4 g + 0.281 g 138.681 g The reason is this: starting at the left-hand side of the calculator answer: The 1 in the hundreds place is definite because it comes from the definite 1 above it.

19. Calculations with significant figures And if these two measurements were added together: 138.4 g + 0.281 g 138.681 g The reason is this: starting at the left-hand side of the calculator answer: The 1 in the hundreds place is definite because it comes from the definite 1 above it.

20. Calculations with significant figures And if these two measurements were added together: 138.4 g + 0.281 g 138.681 g The same holds true for the 3 in the tens place.

21. Calculations with significant figures And if these two measurements were added together: 138.4 g + 0.281 g 138.681 g The same holds true for the 3 in the tens place.

22. Calculations with significant figures And if these two measurements were added together: 138.4 g + 0.281 g 138.681 g The same holds true for the 3 in the tens place. And for the 8 in the ones place.

23. Calculations with significant figures And if these two measurements were added together: 138.4 g + 0.281 g 138.681 g The same holds true for the 3 in the tens place. And for the 8 in the ones place.

24. Calculations with significant figures And if these two measurements were added together: 138.4 g + 0.281 g 138.681 g The same holds true for the 3 in the tens place. And for the 8 in the ones place. But think about the 6 in the tenths place…

25. Calculations with significant figures And if these two measurements were added together: 138.4 g + 0.281 g 138.681 g The same holds true for the 3 in the tens place. And for the 8 in the ones place. But think about the 6 in the tenths place… It came from adding a definite 2 from 0. 2 81 to a guessed 4 from 138. 4.

26. Calculations with significant figures And if these two measurements were added together: 138.4 g + 0.281 g 138.681 g The same holds true for the 3 in the tens place. And for the 8 in the ones place. But think about the 6 in the tenths place… It came from adding a definite 2 from 0. 2 81 to a guessed 4 from 138. 4. If the 4 could have been read as a 3 instead, that would have changed the 6 to a 5.

27. Calculations with significant figures And if these two measurements were added together: 138.3 g + 0.281 g 138.581 g The same holds true for the 3 in the tens place. And for the 8 in the ones place. But think about the 6 in the tenths place… It came from adding a definite 2 from 0. 2 81 to a guessed 4 from 138. 4. If the 4 could have been read as a 3 instead, that would have changed the 6 to a 5.

28. Calculations with significant figures And if these two measurements were added together: 138.4 g + 0.281 g 138.681 g The same holds true for the 3 in the tens place. And for the 8 in the ones place. But think about the 6 in the tenths place… It came from adding a definite 2 from 0. 2 81 to a guessed 4 from 138. 4. If the 4 could have been read as a 3 instead, that would have changed the 6 to a 5. This 6 therefore cannot be considered definite.

29. Calculations with significant figures And if these two measurements were added together: 138.4 g + 0.281 g 138.681 g The same holds true for the 3 in the tens place. And for the 8 in the ones place. But think about the 6 in the tenths place… It came from adding a definite 2 from 0. 2 81 to a guessed 4 from 138. 4. If the 4 could have been read as a 3 instead, that would have changed the 6 to a 5. This 6 therefore cannot be considered definite. And if the 6 is indefinite (guessed), then the answer has to end there.

30. Calculations with significant figures And if these two measurements were added together: 138.4 g + 0.281 g 138.681 g Since the 6 is followed by an 8, it rounds up, giving us…

31. Calculations with significant figures And if these two measurements were added together: 138.4 g + 0.281 g 138.7 g Since the 6 is followed by an 8, it rounds up, giving us…

32. Calculations with significant figures Lets try another problem: 315,600 cm + 219 cm

33. Calculations with significant figures Lets try another problem: 315,600 cm + 219 cm 315,819 cm

34. Calculations with significant figures Lets try another problem: 315,600 cm + 219 cm 315,819 cm This is the calculator answer,

35. Calculations with significant figures Lets try another problem: 315,600 cm + 219 cm 315,819 cm This is the calculator answer, but lets consider where the guesses are.

36. Calculations with significant figures Lets try another problem: 315,600 cm + 219 cm 315,819 cm This is the calculator answer, but lets consider where the guesses are. The 6 in 315,600 cm is in the hundreds place and the 9 in 219 cm is in the ones place.

37. Calculations with significant figures Lets try another problem: 315,600 cm + 219 cm 315,819 cm This is the calculator answer, but lets consider where the guesses are. The 6 in 315,600 cm is in the hundreds place and the 9 in 219 cm is in the ones place. Since we have to round the answer off at the highest guessed place (the hundreds place),

38. Calculations with significant figures Lets try another problem: 315,600 cm + 219 cm 315,819 cm This is the calculator answer, but lets consider where the guesses are. The 6 in 315,600 cm is in the hundreds place and the 9 in 219 cm is in the ones place. Since we have to round the answer off at the highest guessed place (the hundreds place), the answer becomes…

39. Calculations with significant figures Lets try another problem: 315,600 cm + 219 cm 315,800 cm This is the calculator answer, but lets consider where the guesses are. The 6 in 315,600 cm is in the hundreds place and the 9 in 219 cm is in the ones place. Since we have to round the answer off at the highest guessed place (the hundreds place), the answer becomes…

40. Calculations with significant figures Lets try another problem: 34.75 s 0.0248 s + 112.6 s

41. Calculations with significant figures Lets try another problem: 34.75 s 0.0248 s + 112.6 s The fact that there are three numbers being added together does not change anything.

42. Calculations with significant figures Lets try another problem: 34.75 s 0.0248 s + 112.6 s 147.3748 s

43. Calculations with significant figures Lets try another problem: 34.75 s 0.0248 s + 112.6 s 147.3748 s Again, this is the calculator answer,

44. Calculations with significant figures Lets try another problem: 34.75 s 0.0248 s + 112.6 s 147.3748 s Again, this is the calculator answer, but lets consider the guesses:

45. Calculations with significant figures Lets try another problem: 34.75 s 0.0248 s + 112.6 s 147.3748 s Again, this is the calculator answer, but lets consider the guesses: The 5 (in the hundredths place), the 8 (in the thousandths place) and the 6 (in the tenths place) are the three guesses.

46. Calculations with significant figures Lets try another problem: 34.75 s 0.0248 s + 112.6 s 147.3748 s Again, this is the calculator answer, but lets consider the guesses: The 5 (in the hundredths place), the 8 (in the thousandths place) and the 6 (in the tenths place) are the three guesses. The highest guessed place is the tenths place so this is where the answer needs to be rounded.

47. Calculations with significant figures Lets try another problem: 34.75 s 0.0248 s + 112.6 s 147.3748 s Again, this is the calculator answer, but lets consider the guesses: The 5 (in the hundredths place), the 8 (in the thousandths place) and the 6 (in the tenths place) are the three guesses. The highest guessed place is the tenths place so this is where the answer needs to be rounded. This gives and answer of…

48. Calculations with significant figures Lets try another problem: 34.75 s 0.0248 s + 112.6 s 147.4 s Again, this is the calculator answer, but lets consider the guesses: The 5 (in the hundredths place), the 8 (in the thousandths place) and the 6 (in the tenths place) are the three guesses. The highest guessed place is the tenths place so this is where the answer needs to be rounded. This gives and answer of…

49. Calculations with significant figures Lets try word problem:

50. Calculations with significant figures Lets try word problem: A construction crane weighs 197,000 kg.

51. Calculations with significant figures Lets try word problem: A construction crane weighs 197,000 kg. The crane operator sticks his chewed gum under the seat.

52. Calculations with significant figures Lets try word problem: A construction crane weighs 197,000 kg. The crane operator sticks his chewed gum under the seat. The gum weighs 0.038 kg.

53. Calculations with significant figures Lets try word problem: A construction crane weighs 197,000 kg. The crane operator sticks his chewed gum under the seat. The gum weighs 0.038 kg. How much does the crane weigh now?

54. Calculations with significant figures Lets try word problem: A construction crane weighs 197,000 kg. The crane operator sticks his chewed gum under the seat. The gum weighs 0.038 kg. How much does the crane weigh now? 197,000 kg + 0.038 kg The set-up for this problem would look like this:

55. Calculations with significant figures Lets try word problem: A construction crane weighs 197,000 kg. The crane operator sticks his chewed gum under the seat. The gum weighs 0.038 kg. How much does the crane weigh now? 197,000 kg + 0.038 kg The set-up for this problem would look like this: And the calculator answer would be:

56. Calculations with significant figures Lets try word problem: A construction crane weighs 197,000 kg. The crane operator sticks his chewed gum under the seat. The gum weighs 0.038 kg. How much does the crane weigh now? 197,000 kg + 0.038 kg 197,000.038 kg The set-up for this problem would look like this: And the calculator answer would be:

57. Calculations with significant figures Lets try word problem: A construction crane weighs 197,000 kg. The crane operator sticks his chewed gum under the seat. The gum weighs 0.038 kg. How much does the crane weigh now? 197,000 kg + 0.038 kg 197,000.038 kg But lets consider where the guesses are:

58. Calculations with significant figures Lets try word problem: A construction crane weighs 197,000 kg. The crane operator sticks his chewed gum under the seat. The gum weighs 0.038 kg. How much does the crane weigh now? 197,000 kg + 0.038 kg 197,000.038 kg But lets consider where the guesses are: The 7 (in the thousands place), and the 8 (in the thousandths place) are the two guesses.

59. Calculations with significant figures Lets try word problem: A construction crane weighs 197,000 kg. The crane operator sticks his chewed gum under the seat. The gum weighs 0.038 kg. How much does the crane weigh now? 197,000 kg + 0.038 kg 197,000.038 kg But lets consider where the guesses are: The 7 (in the thousands place), and the 8 (in the thousandths place) are the two guesses. The highest guessed place is clearly the thousands place…

60. Calculations with significant figures Lets try word problem: A construction crane weighs 197,000 kg. The crane operator sticks his chewed gum under the seat. The gum weighs 0.038 kg. How much does the crane weigh now? 197,000 kg + 0.038 kg 197,000.038 kg But lets consider where the guesses are: The 7 (in the thousands place), and the 8 (in the thousandths place) are the two guesses. The highest guessed place is clearly the thousands place… So the answer rounds to…

61. Calculations with significant figures Lets try word problem: A construction crane weighs 197,000 kg. The crane operator sticks his chewed gum under the seat. The gum weighs 0.038 kg. How much does the crane weigh now? 197,000 kg + 0.038 kg 197,000 kg But lets consider where the guesses are: The 7 (in the thousands place), and the 8 (in the thousandths place) are the two guesses. The highest guessed place is clearly the thousands place… So the answer rounds to…

62. Calculations with significant figures Lets try word problem: A construction crane weighs 197,000 kg. The crane operator sticks his chewed gum under the seat. The gum weighs 0.038 kg. How much does the crane weigh now? 197,000 kg + 0.038 kg 197,000 kg This might seem strange at first:

63. Calculations with significant figures Lets try word problem: A construction crane weighs 197,000 kg. The crane operator sticks his chewed gum under the seat. The gum weighs 0.038 kg. How much does the crane weigh now? 197,000 kg + 0.038 kg 197,000 kg This might seem strange at first: we started with 197,000 kg, added something to it and ended up with the same mass: 197,000 kg.

64. Calculations with significant figures Lets try word problem: A construction crane weighs 197,000 kg. The crane operator sticks his chewed gum under the seat. The gum weighs 0.038 kg. How much does the crane weigh now? 197,000 kg + 0.038 kg 197,000 kg This might seem strange at first: we started with 197,000 kg, added something to it and ended up with the same mass: 197,000 kg. But this should make sense since what we are adding is so tiny.

65. Calculations with significant figures Lets try word problem: A construction crane weighs 197,000 kg. The crane operator sticks his chewed gum under the seat. The gum weighs 0.038 kg. How much does the crane weigh now? 197,000 kg + 0.038 kg 197,000 kg This might seem strange at first: we started with 197,000 kg, added something to it and ended up with the same mass: 197,000 kg. But this should make sense since what we are adding is so tiny. When a bucket-full of water has one more drop added to it… its still just a bucket-full of water!

66. Calculations with significant figures How about subtraction:

67. Calculations with significant figures How about subtraction: Joeys best time in the 200 m was 21.43 s.

68. Calculations with significant figures How about subtraction: Joeys best time in the 200 m was 21.43 s. He buys new shoes guaranteed to shave 0.2 s off his time.

69. Calculations with significant figures How about subtraction: Joeys best time in the 200 m was 21.43 s. He buys new shoes guaranteed to shave 0.2 s off his time. With the new shoes, what should his best time be?

70. Calculations with significant figures How about subtraction: Joeys best time in the 200 m was 21.43 s. He buys new shoes guaranteed to shave 0.2 s off his time. With the new shoes, what should his best time be? 21.43 s - 0.2 s The set-up for this problem would look like this:

71. Calculations with significant figures How about subtraction: Joeys best time in the 200 m was 21.43 s. He buys new shoes guaranteed to shave 0.2 s off his time. With the new shoes, what should his best time be? 21.43 s - 0.2 s The set-up for this problem would look like this: And the calculator answer would be…

72. Calculations with significant figures How about subtraction: Joeys best time in the 200 m was 21.43 s. He buys new shoes guaranteed to shave 0.2 s off his time. With the new shoes, what should his best time be? 21.43 s - 0.2 s 21.23 s The set-up for this problem would look like this: And the calculator answer would be…

73. Calculations with significant figures How about subtraction: Joeys best time in the 200 m was 21.43 s. He buys new shoes guaranteed to shave 0.2 s off his time. With the new shoes, what should his best time be? 21.43 s - 0.2 s 21.23 s But lets consider where the guesses are:

74. Calculations with significant figures How about subtraction: Joeys best time in the 200 m was 21.43 s. He buys new shoes guaranteed to shave 0.2 s off his time. With the new shoes, what should his best time be? 21.43 s - 0.2 s 21.23 s But lets consider where the guesses are: The 3 (in the hundredths place) and the 2 (in the tenths place) are our two guesses.

75. Calculations with significant figures How about subtraction: Joeys best time in the 200 m was 21.43 s. He buys new shoes guaranteed to shave 0.2 s off his time. With the new shoes, what should his best time be? 21.43 s - 0.2 s 21.23 s But lets consider where the guesses are: The 3 (in the hundredths place) and the 2 (in the tenths place) are our two guesses. The highest guessed place is the tenths place

76. Calculations with significant figures How about subtraction: Joeys best time in the 200 m was 21.43 s. He buys new shoes guaranteed to shave 0.2 s off his time. With the new shoes, what should his best time be? 21.43 s - 0.2 s 21.23 s But lets consider where the guesses are: The 3 (in the hundredths place) and the 2 (in the tenths place) are our two guesses. The highest guessed place is the tenths place so that is where our answer should be rounded. Giving us…

77. Calculations with significant figures How about subtraction: Joeys best time in the 200 m was 21.43 s. He buys new shoes guaranteed to shave 0.2 s off his time. With the new shoes, what should his best time be? 21.43 s - 0.2 s 21.2 s But lets consider where the guesses are: The 3 (in the hundredths place) and the 2 (in the tenths place) are our two guesses. The highest guessed place is the tenths place so that is where our answer should be rounded. Giving us…

78. Calculations with significant figures Now try each of the following 10 problems.

79. Calculations with significant figures Now try each of the following 10 problems. Use a calculator, and then write your answer on paper. Make sure to round the answer at the correct place…

80. Calculations with significant figures Now try each of the following 10 problems. Use a calculator, and then write your answer on paper. Make sure to round the answer at the correct place… …and also make sure to include correct units with each answer.

81. Calculations with significant figures 1) 18.543 cm + 2.61 cm =

82. Calculations with significant figures 1) 18.543 cm + 2.61 cm = 21.15 cm

83. Calculations with significant figures 1) 18.543 cm + 2.61 cm = 21.15 cm The guesses are the 3 (in the thousandths place) and the 1 (in the hundredths place), so the answer must be rounded at the hundredths place.

84. Calculations with significant figures 1) 18.543 cm + 2.61 cm = 21.15 cm Note that the units simply match the units in the two measurements being added. That is always the case for addition and subtraction.

85. Calculations with significant figures 2) 285 g – 17 g =

86. Calculations with significant figures 2) 285 g – 17 g = 268 g

87. Calculations with significant figures 2) 285 g – 17 g = 268 g The guesses are the 5 (in the ones place) and the 7 (also in the ones place), so the answer must be rounded at the ones place.

88. Calculations with significant figures 2) 285 g – 17 g = 268 g Actually, the calculator answer already ended in the ones place, so no rounding was really necessary.

89. Calculations with significant figures 2) 285 g – 17 g = 268 g Actually, the calculator answer already ended in the ones place, so no rounding was really necessary. That happens quite often in addition and subtraction: the calculator answer just happens to be the correct answer!

90. Calculations with significant figures 3) 7.5 m + 11.42 m + 6.3 m =

91. Calculations with significant figures 3) 7.5 m + 11.42 m + 6.3 m = 25.2 m

92. Calculations with significant figures 3) 7.5 m + 11.42 m + 6.3 m = 25.2 m The guesses are the 5 (in the tenths place), the 2 (in the hundredths place) and the 3 (in the tenths place), so the answer must be rounded at the tenths place.

93. Calculations with significant figures 4) 124.58 mg – 122.9 mg =

94. Calculations with significant figures 4) 124.58 mg – 122.9 mg = 1.7 mg

95. Calculations with significant figures 4) 124.58 mg – 122.9 mg = 1.7 mg The guesses are the 8 (in the hundredths place), the 9 (in the tenths place), so the answer must be rounded at the tenths place.

96. Calculations with significant figures 5) 6.45 g + 9.32 g =

97. Calculations with significant figures 5) 6.45 g + 9.32 g = 15.77 g

98. Calculations with significant figures 5) 6.45 g + 9.32 g = 15.77 g The guesses are the 5 (in the hundredths place) and the 2 (also in the hundredths place).

99. Calculations with significant figures 5) 6.45 g + 9.32 g = 15.77 g The guesses are the 5 (in the hundredths place) and the 2 (also in the hundredths place). The calculator answer already ends in the hundredths place so it is correct as it is.

100. Calculations with significant figures 6) 165.4 L – 138.4 L =

101. Calculations with significant figures 6) 165.4 L – 138.4 L = 27.0 L

102. Calculations with significant figures 6) 165.4 L – 138.4 L = 27.0 L The guesses are the 4 (in the tenths place) and the other 4 (also in the tenths place). So the answer should end in the tenths place.

103. Calculations with significant figures 6) 165.4 L – 138.4 L = 27.0 L The guesses are the 4 (in the tenths place) and the other 4 (also in the tenths place). So the answer should end in the tenths place. The calculator just gives 27 as an answer, so here you have to enhance it up to 27.0

104. Calculations with significant figures 7) 130 g + 200 g + 489 g

105. Calculations with significant figures 7) 130 g + 200 g + 489 g 800 g

106. Calculations with significant figures 7) 130 g + 200 g + 489 g 800 g The guesses are the 3 (in the tens place), the 2 (in the hundreds place) and the 9 (in the ones place). So the answer should end in the hundreds place.

107. Calculations with significant figures 8) 130 g + 200 g + 489 g

108. Calculations with significant figures 8) 130 g + 200 g + 489 g 820 g

109. Calculations with significant figures 8) 130 g + 200 g + 489 g 820 g The guesses are the 3 (in the tens place), the 0 (in the tens place) and the 9 (in the ones place). So the answer should end in the tens place this time.

110. Calculations with significant figures 9) 61.7 cm x 9.2 cm =

111. Calculations with significant figures 9) 61.7 cm x 9.2 cm = 570 cm 2

112. Calculations with significant figures 9) 61.7 cm x 9.2 cm = 570 cm 2 This is a multiplication problem, and follows a different rule. Dont look for where the guesses are. Just count how many significant figures there are in each measurement. (Remember???)

113. Calculations with significant figures 9) 61.7 cm x 9.2 cm = 570 cm 2 The 61.7 cm has three significant figures, and the 9.2 cm has just two. Thus, the answer is rounded to just two significant figure.

114. Calculations with significant figures 10) 553 cm + 16.87 cm =

115. Calculations with significant figures 10) 553 cm + 16.87 cm = 570 cm

116. Calculations with significant figures 10) 553 cm + 16.87 cm = 570 cm The guesses are the 3 (in the ones place) and the 7 (in the hundredths place). So the answer should end in the ones place.

117. Calculations with significant figures 10) 553 cm + 16.87 cm = 570 cm The calculator gives an answer of 569.87. The 9 would round up to give an answer of 570, but this appears to end in the tens place. To show the 0 is significant, we put a line over it.

118. Calculations with significant figures Well, hopefully these tutorials have proved helpful in teaching you how to round answers when you are doing calculations with measurements:

119. Calculations with significant figures Well, hopefully these tutorials have proved helpful in teaching you how to round answers when you are doing calculations with measurements: -- When multiplying and dividing, always round your answer off to the fewest number of significant figures.

120. Calculations with significant figures Well, hopefully these tutorials have proved helpful in teaching you how to round answers when you are doing calculations with measurements: -- When multiplying and dividing, always round your answer off to the fewest number of significant figures. -- When adding and subtracting, always round your answer off at the highest guessed place.