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## Arithmetic (all content)

# Multiplying and dividing with significant figures

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

## Want to join the conversation?

- At about8:20, why isn't 10.1 3 sig figs instead of 2?

Because the 0 is sandwiched, it is significant, right?(33 votes)- It is 3 sig figs...he found his mistake and corrected it at8:25. He did the whole equation with it being 3 sig figs but accidently wrote down 2. Haha mistakes happen(13 votes)

- So if your measurements were 103.323 in by 233. in, what would the answer be (without division by 1.07)?(11 votes)
- Since you're dividing a number with 6 sig-figs (103.323 inches) by one with 3 sig-figs (233. inches) your answer would be in 3 sig-figs.(15 votes)

- How do you know how many significant figures to round up to if you have a combination of both multiplication/division and addition/subtraction?(12 votes)
- At8:45into the video, the final answer calculated is 114 tiles but I think that 108 tiles are enough to do the job. Below is my solution. Why am I off by 6 tiles?

If you laid out 11 tiles in a row horizontally, that would span 11.77 feet. That means that 0.30 feet are needed for the last horizontal piece. Now, lay out nine rows, which span 9.63 feet, and the last row needs to span 0.47 feet. You have used up 99 tiles so far.

Now, take 3 tiles and cut them into 3 1.07 by 0.30 sections, use those to span the last column. Then, cut 5 tiles each into two 1.07 by 0.47 sections for the last row. Finally, for the last tile, cut it into one 1.07 by 0.47 section and one 1.07 by 0.30 section.

Total tiles used = 99 + 3 + 5 +1 = 108 tiles(6 votes)- I think you might be making a small mistake in the dimensions of your tile. A tile having an area of 1.07 square feet has dimensions of 1.07 feet x 1 foot, meaning that at any one time, it can cover 1.07 feet along one, not both, of length and breadth.

Thus, 9 rows of 11 tiles each may 11.77 feet lengthwise and thus*not*9.63 but 9 feet breadthwise, leaving*not*0.47 feet to be spanned but 1.1 feet. The left over area at the bottom would be 1.01 x 12.07 = 12.19 sqft. This extra space will mean that you will need 10, not 5 tiles for the last row accounting for 5 of those 6 missing tiles. The last tile is consumed by the 0.1 x 12.07 = 1.207 sqft area at the bottom, accounting for the last missing tile.

9 rows of 11 tiles can also span 9.63 feet breadthwise but only 11 feet lengthwise, thus spanning 0.77 feet less than predicted (leaving 1.07 feet). Thus, you will have to use, not 3 tiles split into threes as predicted, but 11 tiles for the 10.8 sqft space in the last column i.e. 8 tiles more than predicted. The 2 extra tiles in this account are probably due to the fact that I ignored the last tile at the bottom right corner.(10 votes)

- Hey there - can you please do a video on combined operations with significant figures? For example, how would you solve the following problem: (2.756 x 1.20) / (9.5 + 11.28)? Thanks!(7 votes)
- Shouldn't the number of tiles be 113 instead of 114?

How can you round up the number of tiles fitting, the are of tiles fitting will be less than or equal to the area of the floor and cant be greater than the area of the floor.(3 votes)- You are correct, the area of the tiles cannot be greater than the area of the floor. However, we cannot buy 9/10ths of a tile. So, we buy 114 tiles and then cut out whatever area is needed to fill in gaps. [Because the floor wouldn't look too good with a bunch of gaps near the wall. ;)]

Hope this helped!(5 votes)

- So you are saying 5*5=30?(4 votes)
- I think DarkFight is wrong there, at least as far as the 5-5.99 range. If we assume the 5s could have been rounded, then the values they were before hand would have to of been within 4.5 to 5.4 as those are the only values one step of percision greater that could have been rounded to 5. This means the true value could be anywhere from 20.3 to 29.2, which would be one reason why we might want to round to 30 if we could only have 1 significant digit as it is the nearest value of 10 which includes the entire range of potentially more accurate answers.(4 votes)

- I have a similar question as did tiagofischer: from the videos, I understood significant figures as a way to represent the most precise value for a calculation/answer. In the example for the tile-bathroom problem, Mr. Khan rounds up to 114 (8:53). But then why not leave it as 113.9 because it would imply that we were able to measure to the near decimeter as we were when we measured the width of the bathroom (10.1 ft). Thank you and great video!(4 votes)
- For addition and subtraction, we round to the least precise place value. For multiplication and division, however, it is the number of sig figs but not the place value that matters. So for the number 113.9177 etc., you would round to the least number of sig figs in the problem. Both 10.1 and 1.07 have 3 sig figs. Since 113.9177... is a bigger number, you don't even go into decimal places because you can only guarantee that the first three numbers are accurate.(3 votes)

- At8:58Sal said that your precise measurement answer was 114 tiles. Shouldn't it be 114. (with a decimal point) to show that this was not a "rough estimate", but rather a precise calculation?(3 votes)
- If I'm understanding correctly, you only need to do that if the last number(s) is/are 0('s)(3 votes)

- Ok I understand that you round to the lowest number of sig figs in the original problem. So what if I had 2 times 100. The answer would be 200, does that mean that I have to round that to one sig fig? If that is true then the answer would be 2?(2 votes)
- No. It means that only the 2 in the hundreds place will be significant. Significant figures doesn't change place values, only the accuracy that can be expected from them. The two zeros aren't considered significant (in the sense of significant digits/figures)(4 votes)

## 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.