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Current time:0:00Total duration:9:31

Multiplying and dividing with significant figures

Video transcript

now that we have a decent understanding of how to figure out how many significant figures were even dealing with let's think about a situation where we're significant figures will or might become relevant so let's say that I have a carpet here and I using a maybe a meter stick I'm able to measure the carpet to the nearest centimeter and so I get the carpet as on to the nearest centimeter I get to being one point six nine meters so this is this nine obviously this is to the nearest centimeter this nine hundredths of a meter is the same thing as nine centimeters and let's say I'm able to measure the width here the width here as two point zero nine meters I use the same meter stick and you were to ask me Sal what is the area of your carpet and so you know I just do the straight-up calculation the area is just going to be the length times the width so it would be one point six nine meters times two point zero nine meters two point zero nine meters and we could do this by hand but let me just get the the calculator out just to make things move along a little bit faster and so we have one point six nine times two point zero nine and that gives us three point five three two one let me write that down three point five three two one so let me write this in a new color so this gives us three point five three two one and we have a meters times meters which gives us meter squared or square meters and so I might very proudly tell you hey the area here is three point five three to one square meters and the problem here is that it when I give you this thing that has you know all of these numbers behind the decimal point in all of these what we now know to be significant figures it implies that I had a really precise way of measuring of measuring the area what in reality I only was able to measure the area to the nearest centimeter so the way we would do this so that we don't so that I don't make it look like my measurement is more precise and it really is for this calculation that's derived from my measurement 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 three significant figures and over here I have three significant figures three significant figures and so in general when you multiply or divide the significant figures in your product or or or the the product or the I always forget in your there's the devise or there's the dividend then there's the quotient the quotient the the the numbers the significant figures in your product or your quotient cannot be any more than the significant 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 one of these has if this had three significant figures and this had two significant figures I could 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 have to round it to three significant figures and on either so I'm going to round to the nearest hundredth here and so this two will will round down so we go this gets us to 3.5 3.5 3 meters squared and now we're cool with the significant figures let's do another situation with division let's say that I'm let's say that I'm laying tiles down in my bathroom and so the diagram will look very similar and I measure I measure I measure the I measure the width of my bathroom to be let's say it is 10 I'll do it in feet now so let's say it is ten point one ten point one feet and this is the precision that I'm able to measure it with so I'm able to measure to the nearest tenth of a foot and let's say that the length of my floor the length of my floor is let's say the length of my floor I'll just make up a number is twelve point and so for whatever reason I was able to measure this with slightly higher precision so twelve point zero seven feet and let's say let's say that I have tiles I have tiles and the tile has an area so someone else measured it for me it has an area of let's say that the area of this tile is one point zero seven feet squared and this was just measured for me and what 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 floor area is going to be equal to ten point one ten point one feet times twelve point zero seven feet and so that will give us that will give us let's calculate it it is ten point one times twelve point zero seven feet so give us one hundred twenty one point nine zero seven one hundred twenty one point nine zero seven so this is equal to let me scroll over a little bit to the right this is equal to I could do a little bit more to the right this is equal to one hundred and twenty one point nine zero seven 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 okay if this is all you are 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 until you are done especially if you're just doing a mulch 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 the tiles per floor the tiles per 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 one hundred twenty one point nine zero seven feet squared divided by the area of the tile divided by one point zero seven feet squared and once again let me get the calculator out and so we have one twenty one point nine zero seven divided by one point zero seven 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 so it's let me write this in a new color we get one hundred thirteen point nine three one seven seven five seven zero one and it actually just keeps going feet squared and since this is the final answer we're care about how many tiles will fit on to 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 and in general well since we did 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 it clear this isn't two significant figures this is three the one the zero and the one so our final answer can only have three significant figures three significant figures so we need to round to the nearest foot the next that's the next digit over is a nine so we're going to round up so we're going to round up so this will get us to 114 actually this units here aren't in square feet this is in tiles this is feet divided by feet and so this is going to be 114 114 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've 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 you know the minimum number of significant figures in any of the numbers are calculated that's how many significant figures well I guess 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