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Matter and Measurements

**Significant Figures** are often involved in mathematical calculations.

1

Significant Figures Calculations Concept 1

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So now we're gonna see how significant figures can be incorporated in different calculations that will be exposed to in chemistry. Now we're gonna start out with multiplication and division. We're gonna 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 performed the following calculation to the right number of Sig figs. Here we have three values that are being multiplied. Together we have 3.16 times 0.30 to 7 times 5.7 times 10 to the negative three. 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 sick fix for each value from our topic on significant figures. We know that if we have a decimal point, which all of them dio we move from left to right. Now remember, we're going to start counting once you get to our first non zero number. Here three is our first non zero number, and once we start counting, we count all the way into the end. So 123 This has three sick fix for the next one. Skip, Skip, Skip our first non zero is this three? 1234 This has four sig figs. And then finally, we have 5.7 times 10 to the negative three written scientific notation. Remember, when it's written in scientific notation, just focus on the coefficient. We're going to say I'm not our first non zero number is this five? And once we start counting, we count all the way into the end. So one to this has to sick things. Now, based on our sig figs of 34 and two, we have to go with the least number of significant figures. That means our answer at the end can only have two significant figures. So when we first get our answer, what we see initially is 5.452 to to four times 10 to the negative five. We want to sick fix here that for that we have, though we look to the right of it and see if we either keep it is four or we round up next to it. We have this long string of numbers and we have a five there because that number is five. That means we have to round up. So the 54 becomes now and then times 10 to the negative five. 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 scientific notation. So that tells me that my final answer has toe 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.

2

Significant Figures Calculations Concept 2

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now, when either adding, subtracting different numbers. The final answer will contain the least decimal places. If we take a look at this example, it says performed the following calculation to the right number of sick things. Now, if our answer is based on the least number of decimal places, that's gonna have a direct impact on the number of significant figures. If we take a look here, it says we have four or 2.9 minus to 12.2 plus 2.671 If we look at these values, this one has two decimal places. This one here has one decimal place, and this one here has three decimal places. Based on that, we're going with the least number of decimal places are answer can only have one decimal place at the end. When we punch all this in, we get 1 92 561 We can only have one decimal place to the right of that five. There's a six there. That means we have to round up. So this is 1 82 6 as our final answer. And if you wanted to talk about the number of sick figs you'd move from left to right. Our first non zero number is this one and counting all the way through, we'd have four sig figs at the end. By following this least number of decimal places, it has a direct impact on the number of sick figs. 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.

3

Significant Figures Calculations Concept 3

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Now we're dealing with mixed operations, but we have a combination of multiplication, division, subtraction or addition. We're gonna say when dealing with this mixture off multiplication division, addition and subtraction, we must follow the order of operations to help us remember, The order of operations will use pen dis, which stands for parentheses. So what's in parentheses is done first exponents or powers. Then we have multiplication slash division and then addition slash subtraction. So that is our order of operations. Multiplication and division are grouped together. Addition. Subtraction are still group together. If we take a look here, it has performed the following calculation to the right number of sig figs. We have in brackets 1.8 nine times. 10 to the six times 3.5 Then we have 5.21 to the third, divided by 8.8 to 9, minus 6.5 plus 2.920 All right, so we're following our order of operations and in our order of operations, we're gonna do what's in here first because we have brackets and parentheses here. So we're gonna say when we do everything inside of here when we multiply everything. It comes out to But when you're multiplying or dividing numbers, we have to look at the least number of sick fix. So here this number has three sig figs. This number here has four sick fix. So our answer at the end when they multiply, has to have three sig figs. So this initial answer that I got here becomes 5.68 times 10 to the six after I've changed it into scientific notation. Next, we have 5.21 to the third, so that's exponents. So all this means is 5.21 times 5.21 times 5.21 All of them are multiplying each other. All of them have three sig figs. What we would get initially from it is 141. But again, when you're multiplying its least number of sig figs. So our answer would have to have three sig figs. So here, this will come out to be 1 41. Okay, Next we have what's on the bottom here. 8.29 minus 6.5 if you're adding or subtracting its least number of decimal places. So what we would get initially is 2.3 to 9 when we subtract. But this number here has three decimal places. This one here has one decimal place. So our answer at the end must have one decimal place. So that will come out to be 2.3 as our number here next we have plus 2. Okay, Mhm. So we're looking at this portion down here. Now we continue onward now the two numbers on the top on multiplying each other because they're multiplying, it still least sig figs. Here the coefficient has three sig figs and here 1 going the other way because it doesn't have a decimal place. 1 41 has three sig fix. So our answer at the end must have three sig fix when they multiply together comes out as 8.1 times 10 to the eight notice I'm not putting everything all at once. In my calculator you have to do a piece by piece in orderto isolate your final answer. Then on the bottom, these two are adding together. So when they add together. Initially, it comes out as a 5.2 to 0. But when you're adding or subtracting its least number of decimal places, this one here has one. This one here has three. 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 least number of sig figs. This 8.1 has three sig figs in it. This five to has to sig figs in it. So our answer at the end must have to sick fix. So this comes out as 1.5 times 10 to the eight. So this would be our final answer written to the correct number of significant figure 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 is least sig figs, addition or subtraction is least decimal places

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Problem

Perform the following calculation to the right number of sig figs:

[(1.7 × 10^{6}) ÷ (2.63 × 10^{5})] + 6.96

A

13.46

B

13.0

C

13.5

D

14.2

E

4.471

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Problem

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

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Problem

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}

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