Operations with Significant Figures - Online Tutor, Practice Problems & Exam Prep

Understanding significant figures is crucial in mathematical operations involving addition, subtraction, multiplication, and division. For addition and subtraction, round to the least number of decimal places. In multiplication and division, round to the least number of significant figures. When combining operations, follow the order of operations and round to the most significant figures. This ensures precision in calculations, which is essential in scientific contexts, such as determining values in physics and chemistry.

Operations with Significant Figures

Hey, guys. So we've seen how to count up the number of significant figures in our numbers. But a lot of times, you're going to have to do math with sig figs in your problems, like addition, subtraction, multiplication, and division. And so when you're doing math to calculate values, there are 3 additional rules to determine the number of significant figures of your final answer. So I've got these three rules down here, but let's just jump straight into the example so I can show you how they work. So let's check it out. We're going to write the answers expressing the appropriate number of sig figs. The first one is just a simple addition between two numbers.

If you look at these three rules here, which one does it apply to? Well, it turns out this one actually just applies to if it's just addition or subtraction only. Here's the rule: If you're only just doing addition and subtraction, then you're going to round the answer to the same number of decimal places as the number with the least number of decimal places. It's all about the decimal places when you're doing addition and subtraction. For instance, I've got 2 decimal places over here, and then I've got 4 decimal places over here. Plug this straight into your calculator, and we're going to round to the number with the least number of decimal places, which is the second one, which is 2 decimal places. For example, if you plug this into your calculator, you're going to get 8.8501, but we have to round it to the second decimal place. Look at the third digit, if it's 5 or more, we round it up. This rounds to 8.85, and that's the answer.

Let's take a look at the second one. We have 5.83, we have subtraction, and we have addition. But even though this is subtraction and addition, it still falls under the rule of addition and subtraction only. We're only going to round to the decimal places of the number with the least decimal places. Here we've got 3. Here we've got 1. And then here we've got 3. You're going to move from left to right in your calculator. Plug this straight into your calculator exactly as you see it. If you try to do this first, like 4.3 + 103 and then do subtraction, you will get the answer wrong. Always go from left to right. If you plug this into your calculator, you're going to get 1.192. Remember, we have to round now to the number with the least decimal places, which is just the first one. If we round this, now the second digit is a 9, so we round it up. This will be 1.2.

Let's take a look at example C. This one has division and multiplication. This applies to the second rule: Multiplication and division only. The rule is we're going to round the answer to the same number of significant figures as the number with the least number of significant figures. It's not about the decimal places, it's about the number of significant figures. Here, these are all non-zero numbers. This is 6 significant figures. This number, eliminate the leading zeros and count up, has 4 significant figures, and this last number, eliminate the leading zeros, has 2 significant figures. We're going to round to the number with the least number of significant figures, which is this one with 2. Move from left to right in your calculator and plug it in exactly how you see it. You're going to get 1.928274 blah blah blah. But we just have to round it to the second significant figure. It's not about the decimal place. The second significant figure here is actually the 9, so we're going to round to this place. This will be 1.9, and that's our answer.

Now for the last one here, we've got multiplication and addition mixed up inside this number. This falls under the rule of mixing addition and subtraction with multiplication or division. You round the answer to the highest number of significant figures, following the order of operations. Multiplication and division always come before addition and subtraction. If you do this correctly, what you'll get in your calculator is 8.851067. We have to keep track of all the significant figures: 5 here, 2 here, after eliminating the leading zeros, and 3 for this number. The number with the most significant figures is actually this one, so we're going to round it to 5 significant figures. The fifth significant digit is here, so we round this. Looking at the digit beyond our rounding digit, which is higher than 5, we round it up. This makes it 8.85107. And this is actually the answer.

Alright, guys. That's it for this one. Let me know if you have any questions.

What is the area of a sidewalk that is 2.293 m wide and 90 m long? Write your answer with the correct number of significant figures.

• A. 206.4 m²
• B. 210 m²
• C. 206.37 m²
• D. 200 m²

Volume of Two Blocks

Everyone, welcome back. So in this problem, we have these two blocks. We're told some information about the side lengths of block a, which are 0.5 meters, 0.875, and 2.25. And we are directly told what the volume of block b is. We don't have to do any calculations there. We're asked to find what the total combined volume of the blocks is, expressed with the correct number of significant figures. That's really important. Alright? Let's go ahead and get started here.

So, if you want to find the total volume, I'm going to just call this V_total, you can think about this as, well, I've got these two blocks. The total volume of them both combined is really just going to be the volume of block a plus the volume of block b. We already know what the volume of block b is. We just have to figure out what it is for a. And because it's a block, I'm just going to assume that it's a cube, then the volume of a cube is just going to be length times width times height. Right? That's what volume is equal to. So in other words, I can actually write this as one total equation just by doing length times width times height plus the volume of block b.

Now I'm just going to go ahead and replace some of these things with some numbers here. So this is going to be 0.50 times 0.875 times 2.250, and that's going to be plus the volume, which is 2.6. So how do I actually do this? Remember, significant figures are going to be important, especially when you're dealing with calculations involving operations. And if you look here, I actually have a mixture of operations. I've got multiplication here over in this parenthesis, and then I've got addition. So I'm going to stick to rule 3, right, which is that I'm going to have to use PEMDAS, and then I'm going to have to sort of save all the values in the intermediate steps, and I'll do all the rounding at the end. So the first thing I'm going to have to do here is I'm going to have to figure out what is this first intermediate step where I'm only just doing multiplication.

Well, for multiplication, we're going to follow rule 2. Right? When we're doing multiplication, we're looking at the significant figures. Here, there are two significant figures. Here, there's three, and then here, there are four. If I just plug this into my calculator, what I'm going to get is I'm going to get 0.984375. So remember, I'm not going to round yet. I'm going to save all of these numbers. But what I'm going to do is I'm going to indicate where my last significant figure is. I'm going to put a little mark here. So the last one that's significant is the 8 because this is the second significant figure. This is the least number of significant figures that I have.

Alright? So without rounding anything, now what I'm going to do is I'm going to take 0.984375, and I'm going to add this to 2.6. So now what happens is, now you're actually just doing addition and subtraction. So now for this final answer here, we're going to have to stick to the rule where we're going to look at the decimal places, and we're going to have to stick to the least number of decimal places. Alright? So what happens here is, here we have one decimal place. I'm going to put 1 DP here, and this is going to be the second decimal place. That's the last significant figure for that number over here. So when you add these things together, what you're going to get is 3.584375. But now we have to round it. So right? So we're going to round this final answer to the first decimal place, and what you're going to get here is 3.6 meters cubed. Right? So this is the correct answer.

So notice how, again, we don't want to round in the intermediate steps because by rounding, you're basically sacrificing some accuracy, and then your answer actually could be off by quite a bit if you just keep on rounding in the immediate steps. Alright. So it's always good to keep all your digits and then round at the end according to that rule. That's it for this one, folks. Let me know if you have any questions.

Here’s what students ask on this topic:

What are the rules for significant figures in addition and subtraction?

When performing addition or subtraction, the result should be rounded to the same number of decimal places as the number with the least number of decimal places in the original numbers. For example, if you add 12.34 (2 decimal places) and 1.2 (1 decimal place), the result should be rounded to 1 decimal place. This ensures that the precision of the result is consistent with the least precise measurement.

How do you determine significant figures in multiplication and division?

In multiplication and division, the result should be rounded to the same number of significant figures as the number with the least number of significant figures in the original numbers. For instance, if you multiply 4.56 (3 significant figures) by 1.2 (2 significant figures), the result should be rounded to 2 significant figures. This rule ensures that the precision of the result matches the least precise measurement.

What is the rule for significant figures when combining addition/subtraction with multiplication/division?

When combining addition/subtraction with multiplication/division, follow the order of operations (PEMDAS/BODMAS). Perform the multiplication or division first, then the addition or subtraction. Round the intermediate result to the appropriate number of significant figures before proceeding. Finally, round the final result to the most significant figures among the numbers involved. This approach maintains the precision of the calculations throughout the process.

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. If the digit immediately to the right is 5 or greater, round up the identified digit. If it is less than 5, leave the identified digit unchanged. For example, to round 3.456 to 2 significant figures, look at the third digit (6). Since it is greater than 5, round up the second digit (4) to 5, resulting in 3.5.

Why are significant figures important in scientific calculations?

Significant figures are crucial in scientific calculations because they convey the precision of measurements and ensure that results are not over-interpreted. They help maintain consistency and accuracy in reporting data, which is essential in fields like physics and chemistry. By using significant figures, scientists can communicate the reliability of their measurements and calculations, avoiding misleading conclusions based on overly precise or imprecise data.