Significant Figures: In Calculations - Online Tutor, Practice Problems & Exam Prep

In calculations involving significant figures, multiplication and division yield results based on the least number of significant figures, while addition and subtraction depend on the least number of decimal places. For mixed operations, follow the order of operations (PENDES: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction). This ensures accurate results, such as determining significant figures in scientific notation and rounding appropriately. Understanding these principles is crucial for precise measurements in chemistry, impacting calculations involving acids, bases, and various chemical reactions.

Significant Figures Calculations Concept 1

So now we're going to see how significant figures can be incorporated in different calculations that we'll be exposed to in chemistry. Now we're going to start out with multiplication and division. We're going to say when either multiplying or dividing different numbers, the final answer will contain the least significant figures. And if we take a look at this example, it says, perform the following calculation to the right number of sig figs. Here we have 3 values that are being multiplied together. We have 3.16 times 0.003027 times 5.7 times 10^-3. We just said that when you're multiplying or dividing its least number of sig figs for your final answer, so we need to determine the number of sig figs for each value. From our topic on significant figures, we know that if we have a decimal point, which all of them do, we move from left to right. Now remember, we're going to start counting once you get to our first non-zero number. Here, 3 is our first non-zero number, and once we start counting, we count all the way into the end. So 1, 2, 3, this has 3 sig figs. For the next one, skip, skip, skip, our first non-zero is this 3. 1, 2, 3, 4. This has 4 sig figs. And then finally we have 5.7 times 10^-3 written in scientific notation. Remember, when it's written in scientific notation, just focus on the coefficient. We're going to say our first non-zero number is this 5, and once we start counting, we count all the way into the end. So one, 2. This has 2 sig figs.

Now based on our sig figs of 3, 4, and 2, we have to go with the least number of significant figures. That means our answer at the end can only have 2 significant figures. So when we first get our answer, what we see initially is 5.4522324 times 10^-5. We want 2 sig figs here. That 4 that we have though, we look to the right of it and see if we either keep it as 4 or we round up. Next to it, we have this long string of numbers, and we have a 5 there. Because that number is 5, that means we have to round up. So the 5.4 becomes now 5.5, and then times 10^-5. This represents our answer, which has the least number of significant figures based on the initial values given. We were given these three numbers initially and the one with the least number of sig figs was the one written in scientific notation. So that tells me that my final answer has to have that number of significant figures.

Now that we've looked at multiplication and division, let's go on to our next video and let's see what happens when we incorporate addition and subtraction.

Significant Figures Calculations Concept 2

Now when either adding or subtracting different numbers, the final answer will contain the least decimal places. If we take a look at this example, it says perform the following calculation to the right number of significant figures. Now, if our answer is based on the least number of decimal places, that's going to have a direct impact on the number of significant figures. If we take a look here, it says we have 402.09 minus 212.2 plus 2.671. If we look at these values, this one has 2 decimal places, this one here has 1 decimal place, and this one here has 3 decimal places. Based on that, we're going with the least number of decimal places, our answer can only have 1 decimal place at the end.

When we punch all this in, we get 192.561. We can only have 1 decimal place. To the right of that 5, there's a 6 there. That means we have to round up. So this is 182.6 as our final answer, and if you wanted to talk about the number of significant figures, you'd move from left to right. Our first non-zero number is this 1, and counting all the way through, we'd have 4 significant figures at the end.

By following this rule of the least number of decimal places, it has a direct impact on the number of significant figures in our final answer. Up to this point, we've kept multiplication and division separate from addition and subtraction. But what happens when you mix them together? To find out what to do, click on the next video.

Significant Figures Calculations Concept 3

Now we're dealing with mixed operations. We have a combination of multiplication, division, subtraction, or addition. We're going to say when dealing with this mixture of multiplication, division, addition, and subtraction, we must follow the order of operations. To help us remember the order of operations, we use PENDES, which stands for parentheses, so what's in parentheses is done first, exponents or powers, then we have multiplication/division, and then addition/subtraction. So that is our order of operations. Multiplication and division are grouped together, addition and subtraction are still grouped together. If we take a look here, it says perform the following calculation to the right number of significant figures.

We have in brackets: 1.89 × 10 6 × 3.005 . Then we have 5.21 3 / ( 8.829 - 6.5 ) + 2.920 . Alright. So we're following our order of operations, and in our order of operations, we're going to do what's in here first because we have brackets and parentheses here. So we're going to say, when we do everything inside of there, when we multiply everything, it comes out to 5679450. But when you're multiplying or dividing numbers, we have to look at the least number of significant figures.

So here, this number has 3 significant figures, this number here has 4 significant figures. So our answer at the end when they multiply has to have 3 significant figures. So this initial answer that I got here becomes 5.68 × 10 6 , after I've changed it into scientific notation.

Next, we have 5.21 3 , so that's exponent. So all this means is 5.21 × 5.21 × 5.21 . All of them are multiplying each other, all of them have 3 significant figures. What we would get initially from it is 141.420761. But again, when you're multiplying, the least number of significant figures, so our answer would have to have 3 significant figures. So here, this will come out to be 141.

Next, we have what's on the bottom here, 8.829 minus 6.5. If you're adding or subtracting it's the least number of decimal places, so what we would get initially is 2.329 when we subtract. But this number here has 3 decimal places, this one here has 1 decimal place. So our answer at the end must have 1 decimal place. So that will come out to be 2.3 as our number here. Next, we have plus 2.920. So we're looking at this portion down here now.

We continue onward. Now the two numbers on the top are multiplying each other. Because they're multiplying, it's still least significant figures. Here, the coefficient has 3 significant figures, and here 141, going the other way because it doesn't have a decimal place, 141 has 3 significant figures. So our answer at the end must have 3 significant figures. When they multiply together it comes out as 8.01 × 10 8 . Notice I'm not putting everything all at once in my calculator. You have to do it piece by piece in order to isolate your final answer.

Then on the bottom, these 2 are adding together, so when they add together initially, it comes up as, 5.220. But when you're adding or subtracting, it's the least number of decimal places. This one here has 1, this one here has 3, so your answer at the end must have one decimal place. So that would be 5.2. Now we just have these two numbers that are dividing each other, so again it's the least number of significant figures. This 8.01 has 3 significant figures in it. This 5.2 has 2 significant figures in it. So our answer at the end must have 2 significant figures. So this comes out as 1.5 × 10 8 . So this would be our final answer written to the correct number of significant figures based on this mix of operations.

So just keep in mind the order of operations to guide you on what to do. And remember, multiplication and division are the least significant figures, addition or subtraction is the least decimal places.

Perform the following calculation to the right number of sig figs:
[(1.7 × 10⁶) ÷ (2.63 × 10⁵)] + 6.96

• A. 13.46
• B. 13.0
• C. 13.5
• D. 14.2
• E. 4.471

Perform the following mathematical operations and express the result to the correct number of significant figures.
(6.404 × 2.91) / (18.7 – 17.1)

• A. 11.65
• B. 11.7
• C. 10
• D. 12
• E. 11.625

What answer should be reported, with the correct number of significant figures, for the following calculation?
[(42.00 − 40.914) ⋅ (25.739 − 25.729)] / [(11.50⋅1.001) + (0.00710 ⋅ 700.)]

• A. 7.0 x 10^-4
• B. 6.67 x 10^-4
• C. 6.70 x 10^-4
• D. 6.7 x 10^-4
• E. 6.75 x 10^-4

Here’s what students ask on this topic:

How do you determine the number of significant figures in a multiplication or division calculation?

When performing multiplication or division, the final answer should have the same number of significant figures as the value with the least number of significant figures in the calculation. For example, if you multiply 3.16 (3 significant figures) by 0.003027 (4 significant figures) and 5.7 × 10^-3 (2 significant figures), the result should be rounded to 2 significant figures, as 2 is the smallest number of significant figures among the values.

How do you handle significant figures in addition and subtraction?

In addition and subtraction, the final answer should be rounded to the least number of decimal places present in any of the numbers being added or subtracted. For instance, if you have 402.09 (2 decimal places), 2.1 (1 decimal place), and 2.671 (3 decimal places), the result should be rounded to 1 decimal place, as that is the smallest number of decimal places among the values.

What is the order of operations for mixed calculations involving significant figures?

For mixed operations involving significant figures, follow the order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction (PEMDAS). Perform calculations inside parentheses first, then handle exponents, followed by multiplication and division, and finally addition and subtraction. Apply the rules for significant figures at each step: least number of significant figures for multiplication/division and least number of decimal places for addition/subtraction.

How do you round numbers to the correct number of significant figures?

To round a number to the correct number of significant figures, identify the digit at the desired place and look at the digit immediately to its right. If this digit is 5 or greater, round up the target digit. If it is less than 5, keep the target digit the same. For example, to round 5.4522324 to 2 significant figures, look at the third digit (4). Since the next digit (5) is 5 or greater, round up to 5.5.

How do significant figures affect scientific notation?

When using scientific notation, the number of significant figures is determined by the coefficient (the number before the exponent). For example, in 5.7 × 10^-3, the coefficient 5.7 has 2 significant figures. When performing calculations, ensure the final result is expressed with the correct number of significant figures based on the rules for multiplication, division, addition, or subtraction.