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?

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.)

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) 10 20 30 40 50 cm

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). 10 cm 20

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

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 cm2. But if we state the area to be 243.21cm2, we are stating a pretty high level of precision.

Calculations with significant figures But if we say the area is 243.21cm2, we are saying we know the area to a very high level of precision.

Calculations with significant figures But if we say the area is 243.21cm2, 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.”

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 cm2 would have to be a correct area for the same rectangle.

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 cm2 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.”

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 cm2 would also have to be a correct area for the same rectangle.

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 cm2 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.”

Calculations with significant figures 243.21 cm2, 238.92 cm2 and 247.52 cm2

Calculations with significant figures 243.21 cm2, 238.92 cm2 and 247.52 cm2 These three values cannot all be correct.

Calculations with significant figures 243.21 cm2, 238.92 cm2 and 247.52 cm2 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.

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.

Calculations with significant figures 243.21 cm2 rounds to 240 cm2 238.92 cm2 rounds to 240 cm2 and 247.52 cm2 rounds to 250 cm2.

Calculations with significant figures 240 cm2, 240 cm2 and 250 cm2.

Calculations with significant figures 240 cm2, 240 cm2 and 250 cm2. These are all consistent with one another.

Calculations with significant figures 240 cm2, 240 cm2 and 250 cm2. These are all consistent with one another. They all have two significant figures, and they show disagreement only in the guessed digit.

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?

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.

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

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

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

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

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…

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

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

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.

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

Calculations with significant figures The calculator answer has 6 significant figures. 23.40 cm x 0.47 cm x 6.05 cm = 66.5379 cm3 precise to 4 significant figures precise to 2 significant figures precise to 3 significant figures

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 cm3 precise to 4 significant figures precise to 2 significant figures precise to 3 significant figures

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: 67 cm3 23.40 cm x 0.47 cm x 6.05 cm = 66.5379 cm3 precise to 4 significant figures precise to 2 significant figures precise to 3 significant figures

Calculations with significant figures When you report an answer to be something like “66.5379 cm3”(just because that is what showed up on your calculator), you are claiming a level of precision much higher than the measurements deserve. 67 cm3 23.40 cm x 0.47 cm x 6.05 cm = 66.5379 cm3 precise to 4 significant figures precise to 2 significant figures precise to 3 significant figures

Calculations with significant figures An answer of 66.5379 cm3 means that the “66.537..” are definite, and only the “9” is a guess. 67 cm3 23.40 cm x 0.47 cm x 6.05 cm = 66.5379 cm3 precise to 4 significant figures precise to 2 significant figures precise to 3 significant figures

Calculations with significant figures An answer of 66.5379 cm3 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. 67 cm3 23.40 cm x 0.47 cm x 6.05 cm = 66.5379 cm3 precise to 4 significant figures precise to 2 significant figures precise to 3 significant figures

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)

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.

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.

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

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.

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.

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. 100 200 300 400 500 600 700 800 m

Calculations with significant figures Speed = = = distance time 680 m 146.39 s 100 200 300 400 500 600 700 800 m

Calculations with significant figures Speed = = = 4.645126033 m/s distance time 680 m 146.39 s 100 200 300 400 500 600 700 800 m

Calculations with significant figures Speed = = = 4.645126033 m/s This answer is what appears on the calculator, but it is obviously way too precise. distance time 680 m 146.39 s 100 200 300 400 500 600 700 800 m

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? distance time 680 m 146.39 s 100 200 300 400 500 600 700 800 m

Calculations with significant figures Speed = = = 4.645126033 m/s The distance (680 m) has two significant figures distance time 680 m 146.39 s 100 200 300 400 500 600 700 800 m

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. distance time 680 m 146.39 s 100 200 300 400 500 600 700 800 m

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... distance time 680 m 146.39 s 100 200 300 400 500 600 700 800 m

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. distance time 680 m 146.39 s 100 200 300 400 500 600 700 800 m

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. 4.6 m/s distance time 680 m 146.39 s 100 200 300 400 500 600 700 800 m

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

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

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…

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.

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

Calculations with significant figures 1) 34.6 m x 14.8 m = 512 m2

Calculations with significant figures 1) 34.6 m x 14.8 m = 512 m2 The 34.6 m and the 14.8 m both have three significant figures, so the answer is rounded to three significant figures.

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

Calculations with significant figures 2) 67 cm x 38 cm = 2500 cm2

Calculations with significant figures 2) 67 cm x 38 cm = 2500 cm2 The 67 cm and the 38 cm both have two significant figures, so the answer must be rounded to two significant figures.

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

Calculations with significant figures 3) 0.0057 m x 0.0063 m = 0.000036 m2

Calculations with significant figures 3) 0.0057 m x 0.0063 m = 0.000036 m2 The 0.0057 m and the 0.0063 m both have two significant figures, so the answer must be rounded to two significant figures.

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

Calculations with significant figures 4) 34.0 mm x 72.7 mm = 2470 mm2

Calculations with significant figures 4) 34.0 mm x 72.7 mm = 2470 mm2 The 34.0 mm and the 72.7 mm both have three significant figures, so the answer is rounded to three significant figures.

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

Calculations with significant figures 5) 18.38 m x 23.54 m = 432.7 m2

Calculations with significant figures 5) 18.38 m x 23.54 m = 432.7 m2 The 18.38 m and the 23.54 m both have four significant figures, so the answer is rounded to four significant figures.

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

Calculations with significant figures 6) 207 m x 64 m = 13,000 m2

Calculations with significant figures 6) 207 m x 64 m = 13,000 m2 The 207 m has three significant figures, but the 64 m has only two. Thus, the answer should be rounded to two significant figures.

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

Calculations with significant figures 7) 296.3 cm x 0.8 cm = 200 cm2

Calculations with significant figures 7) 296.3 cm x 0.8 cm = 200 cm2 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.

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

Calculations with significant figures 8) 3.4 m x 16.3 m x 25.7 m = 1400 m3

Calculations with significant figures 8) 3.4 m x 16.3 m x 25.7 m = 1400 m3 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.

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

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

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.

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

Calculations with significant figures 10) 25.50 m x 12.00 m = 306.0 m2

Calculations with significant figures 10) 25.50 m x 12.00 m = 306.0 m2 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”

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

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

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.

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

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

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.

Calculations with significant figures 13) 2300 m2 - 725 m =

Calculations with significant figures 13) 2300 m2 - 725 m = 3.2 m

Calculations with significant figures 13) 2300 m2 - 725 m = 3.2 m The 2300 m2 has two significant figures and the 725 m has three. The answer should therefore be rounded to just two significant figures.

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

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

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.”

Calculations with significant figures 15) 32.0 m3 – 7.0 m =

Calculations with significant figures 15) 32.0 m3 – 7.0 m = 4.6 m2

Calculations with significant figures 15) 32.0 m3 – 7.0 m = 4.6 m2 The 32.0 m3 has three significant figures, but the 7.0 m has only two, so the answer should be rounded to just two significant figures.

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

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

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.

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

Calculations with significant figures 17) (65 m x 17 m) – 4.83 s = 230 m2/s

Calculations with significant figures 17) (65 m x 17 m) – 4.83 s = 230 m2/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.

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

Calculations with significant figures 18) 56.94 g – (3.42 cm x 7.61 cm x 0.35 cm) = 6.3 g/cm3

Calculations with significant figures 18) 56.94 g – (3.42 cm x 7.61 cm x 0.35 cm) = 6.3 g/cm3 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.

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

Calculations with significant figures 19) 215 cm x 372 cm = 80,000 cm2

Calculations with significant figures 19) 215 cm x 372 cm = 80,000 cm2 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.

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

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

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.

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

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

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.

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...

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.

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

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

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).

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?

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.

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

Calculations with significant figures In adding and subtracting, you still look for the weakest measurement, but weak is defined differently. You don’t count significant figures at all.

Calculations with significant figures In adding and subtracting, you still look for the weakest measurement, but weak is defined differently. You don’t count significant figures at all. Instead, you look at what place the guess (the last significant figure) is in.

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

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. 1 , . ten-thousands place thousands place hundreds place tens place ones place tenths place hundredths place thousandths place ten-thousandths place

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. 1 , . ten-thousands place thousands place hundreds place tens place ones place tenths place hundredths place thousandths place ten-thousandths place

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

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.

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) .

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…

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…

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:

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.

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.

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.

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.

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.

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.

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…

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 0.281 to a guessed “4” from 138.4.

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 0.281 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.”

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 0.281 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.”

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 0.281 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.

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 0.281 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.

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…

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…

Calculations with significant figures Let’s try another problem: 315,600 cm + 219 cm

Calculations with significant figures Let’s try another problem: 315,600 cm + 219 cm 315,819 cm

Calculations with significant figures Let’s try another problem: 315,600 cm + 219 cm 315,819 cm This is the calculator answer,

Calculations with significant figures Let’s try another problem: 315,600 cm + 219 cm 315,819 cm This is the calculator answer, but let’s consider where the guesses are.

Calculations with significant figures Let’s try another problem: 315,600 cm + 219 cm 315,819 cm This is the calculator answer, but let’s 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.

Calculations with significant figures Let’s try another problem: 315,600 cm + 219 cm 315,819 cm This is the calculator answer, but let’s 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),

Calculations with significant figures Let’s try another problem: 315,600 cm + 219 cm 315,819 cm This is the calculator answer, but let’s 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…

Calculations with significant figures Let’s try another problem: 315,600 cm + 219 cm 315,800 cm This is the calculator answer, but let’s 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…

Calculations with significant figures Let’s try another problem: 34.75 s 0.0248 s + 112.6 s

Calculations with significant figures Let’s 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.

Calculations with significant figures Let’s try another problem: 34.75 s 0.0248 s + 112.6 s 147.3748 s

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

Calculations with significant figures Let’s try another problem: 34.75 s 0.0248 s + 112.6 s 147.3748 s Again, this is the calculator answer, but let’s consider the guesses:

Calculations with significant figures Let’s try another problem: 34.75 s 0.0248 s + 112.6 s 147.3748 s Again, this is the calculator answer, but let’s 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.

Calculations with significant figures Let’s try another problem: 34.75 s 0.0248 s + 112.6 s 147.3748 s Again, this is the calculator answer, but let’s 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.

Calculations with significant figures Let’s try another problem: 34.75 s 0.0248 s + 112.6 s 147.3748 s Again, this is the calculator answer, but let’s 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…

Calculations with significant figures Let’s try another problem: 34.75 s 0.0248 s + 112.6 s 147.4 s Again, this is the calculator answer, but let’s 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…

Calculations with significant figures Let’s try word problem:

Calculations with significant figures Let’s try word problem: A construction crane weighs 197,000 kg.

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

Calculations with significant figures Let’s 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.

Calculations with significant figures Let’s 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?

Calculations with significant figures Let’s 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:

Calculations with significant figures Let’s 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:

Calculations with significant figures Let’s 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:

Calculations with significant figures Let’s 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 let’s consider where the guesses are:

Calculations with significant figures Let’s 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 let’s consider where the guesses are: The “7” (in the thousands place), and the “8” (in the thousandths place) are the two guesses.

Calculations with significant figures Let’s 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 let’s 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…

Calculations with significant figures Let’s 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 let’s 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…

Calculations with significant figures Let’s 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 But let’s 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…

Calculations with significant figures Let’s 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 This might seem strange at first:

Calculations with significant figures Let’s 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 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.

Calculations with significant figures Let’s 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 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.

Calculations with significant figures Let’s 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 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… it’s still just a bucket-full of water!

Calculations with significant figures How about subtraction:

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

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

Calculations with significant figures How about subtraction: Joey’s 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?

Calculations with significant figures How about subtraction: Joey’s 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:

Calculations with significant figures How about subtraction: Joey’s 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…

Calculations with significant figures How about subtraction: Joey’s 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…

Calculations with significant figures How about subtraction: Joey’s 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 let’s consider where the guesses are:

Calculations with significant figures How about subtraction: Joey’s 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 let’s consider where the guesses are: The “3” (in the hundredths place) and the “2” (in the tenths place) are our two guesses.

Calculations with significant figures How about subtraction: Joey’s 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 let’s 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

Calculations with significant figures How about subtraction: Joey’s 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 let’s 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…

Calculations with significant figures How about subtraction: Joey’s 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 let’s 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…

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

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…

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.

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

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

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.

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.

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

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

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.

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.

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!

Calculations with significant figures 7.5 m + 11.42 m + 6.3 m =

Calculations with significant figures 7.5 m + 11.42 m + 6.3 m = 25.2 m

Calculations with significant figures 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.

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

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

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.

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

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

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).

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.

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

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

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.

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”

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

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

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.

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

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

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.

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

Calculations with significant figures 9) 61.7 cm x 9.2 cm = 570 cm2

Calculations with significant figures 9) 61.7 cm x 9.2 cm = 570 cm2 This is a multiplication problem, and follows a different rule. Don’t look for where the guesses are. Just count how many significant figures there are in each measurement. (Remember???)

Calculations with significant figures 9) 61.7 cm x 9.2 cm = 570 cm2 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.

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

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

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.

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.

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:

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.

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.