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Multiplying and dividing with significant figures
Now that we have a decent understanding of how to figure out how many significant figures we're even dealing with, let's think about a situation where significant figures will or might become relevant. So let's say that I have a carpet here. And using maybe a meter stick, I'm able to measure the carpet to the nearest centimeter. And so I get the carpet as-- to the nearest centimeter, I get it to being 1.69 meters. So this 9, obviously, this is to the nearest centimeter. This 9/100 of a meter is the same thing as 9 centimeters. And let's say I'm able to measure the width here as 2.09 meters. I use the same meter stick. And you were to ask me, Sal, what is the area of your carpet? And so I'd just do the straight-up calculation. The area is just going to be the length times the width, so it would be 1.69 meters times 2.09 meters. And we could do this by hand, but let me just get the calculator out just to make things move along a little bit faster. And so we have 1.69 times 2.09. And that gives us 3.5321. Let me write that down-- 3.5321. So let me write this in a new color. So this gives us 3.5321, and we have a meters times a meters, which gives us meters squared or square meters. And so I might very proudly tell you, hey, the area here is 3.5321 square meters. And the problem here is that when I give you this thing that has all of these numbers behind the decimal point and all of these what we now know to be significant figures, it implies that I had a really precise way of measuring the area. Well, in reality, I only was able to measure the area to the nearest centimeter. So the way we would do this so that I don't make it look like my measurement is more precise than it really is. Or this calculation that's derived from my measurements-- I make sure that it has no more significant figures than either of the numbers that I multiplied. So in this situation, I have three significant figures here. And over here, I have three significant figures. And so in general, when you multiply or divide, the significant figures in your product or-- I always forget. There's the divisor, there's the dividend, and there's the quotient. The significant figures in your product or your quotient cannot be any more than the least number of significant digits in whatever you are using to come up with that product or quotient. So over here, both of these have three significant figures. So I can only have three significant figures in my product. If this had three significant figures and this had two significant figures, I could have only have two significant figures in my product. So in order to be kind of legit here, I have to round this to three significant figures. So I'm going to round to the nearest hundredth here. And so this 2 we'll round down. So this gets us to 3.53 meters squared. And now we're cool with the significant figures. Let's do another situation with division. Let's say that I'm laying tiles down in my bathroom. And so the diagram will look very similar. And I measure the width of my bathroom to be-- I'll do it in feet, now. So let's say it is 10.1 feet. And this is the precision that I'm able to measure it with. So I'm able to measure it to the nearest tenth of a foot. And let's say that the length of my floor-- I'll just make up a number-- is 12 point-- and for whatever reason, I was able to measure this with slightly higher precision, so 12.07 feet. And let's say that I have tiles. And the tile has an area-- so someone else measured it for me. Let's say that the area of this tile is 1.07 feet squared. And this was just measured for me. And I want to do is I want to figure out how many tiles can fit on this bathroom floor. So what I would do is I would figure out the area of this bathroom floor and then divide by the area of the tiles. And so the area of the bathroom floor, so floor area is going to be equal to 10.1 feet times 12.07 feet. And so that will give us-- let's calculate it. It is 10.1 times 12.07 feet. So it gives us 121.907. So this is equal to-- let me scroll over a little bit to the right. I could go a little bit more to the right. This is equal to 121.907 feet squared, or square feet. Now, we're not done with our calculation. But there might be a temptation right here to say, look, I had four significant figures over here. I have three significant figures over here. There would be a temptation to say, look, my area should not have more than three significant figures. And that temptation would be OK if this is all you were looking for, if the final answer you were looking for was the area of the floor. But we're not done with our calculation. We want to figure out how many of these tiles will fit into this area. And so the general rule of thumb-- because you don't want to lose information. The general rule of thumb is don't round to significant figures until you are done with your calculation, especially if you're just doing a bunch of multiplying and dividing. Because otherwise, if you round here, you actually will introduce more error into your calculation than you'd want to. So what you do is you keep it as kind of the full number. Now you do the division. So let's do the division. So the tiles per floor-- I guess we could say my bathroom, or tiles in the bathroom, tiles fitting in bathroom, in the floor of this bathroom. It would be the area of the bathroom, so 121.907 feet squared, divided by the area of the tile, divided by 1.07 feet squared. And once again, let me get the calculator out. And so we have 121.907 divided by 1.07. And you get this crazy thing with all of these digits. But this is going to be our final answer, so here we do care about significant figures. So tiles fitting in the bathroom, we get something that actually just keeps going. Let me write this in a new color. We get 113.931775701-- and it actually just keeps going-- feet squared. And since this is the final answer-- we care about how many tiles will fit onto this bathroom floor-- now the significant figures come into play. And the way to think about this is I have four significant figures over here. I have two significant figures over here. I have three significant figures over here. And since we did just a bunch of multiplying and dividing, we have to have the minimum. Whatever is the minimum significant figures of the things that we computed with, that's how many significant figures we can have in our final answer. Oh, and let me make this clear. This isn't two significant figures, this is three-- the 1, the 0, and the 1. So our final answer can only have three significant figures. So we need to round to the nearest foot. The next digit over is a 9, so we're going to round up. So this will get us to 114. Actually, the units here aren't in square feet. This is in tiles. This is feet divided by feet, and so this is going to be 114 tiles. Obviously, it's not going to be exactly 114 tiles. But based on the precision of the measurements we've done, we can say 114 tiles. Now, what I have just showed you right here is when we multiply and divide measurements that have a certain number of significant figures. The general rule of thumb is whatever is the minimum number of significant figures in any of the numbers you've calculated, that's how many significant figures-- or the least number is the number of significant figures in your final quotient or product or answer. When you do addition and subtraction, it's a little bit different. And we'll cover that in the next video.