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# Multiplying and dividing with significant figures

Learn to multiply and divide with significant figures. Created by Sal Khan.

Video transcript

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.