Hi! Welcome to Math Antics. In this lesson, we are gonna learn about long division.
If you haven’t already watched our video about basic division, then be sure to go back and watch that first.
It will make learning long division a lot easier.
Long division is just a way of breaking up a bigger division problem into a series of short division steps
like the ones that we did in the basic division video.
The nice thing about long division is that once you know the procedure,
you can divide up all kinds of numbers, even if they are REALLY big.
The key to long division is to think about our division problem digit-by-digit.
If our dividend (the number we’re dividing up) has a lot of digits,
then that means that there will be a lot of division steps to do.
When we learned basic one-step division, all of the dividends were small enough
that we could just use the multiplication table to help us find the answer.
But what if we have a division problem like this? 936 divided by 4
936 is definitely NOT on our multiplication table!
In fact, there’s not anything even close to 936, so what do we do?
Well, instead of trying to divide the entire 936 by 4 all at once,
let’s break this problem up into smaller steps by just trying to divide each digit by 4,
one digit at a time …digit-by-digit.
Do you remember how with multi-digit multiplication and addition,
we always start with the smallest digit (the ones place digit) and we work from right to left?
Well division is backwards! We still go digit by digit, but the other way.
We start by trying to divide up the digit in the biggest number place first and we work our way from left to right.
So the first step in this problem is to divide the FIRST digit of our dividend by 4.
We’ll just ignore the other digits for now, and that makes it look like we have the division problem 9 divided by 4.
Great! That’s easy! It’s just a basic division problem like in the last video.
So we ask, “How many ‘4’s will it take to make 9 or almost 9?”
Well, two ‘4’s would be 8, and that’s almost 9.
So just like before, we put the 2 in our answer spot on top of the line.
But wait a minute… there’s a lot of room up there. Where exactly do we put it?
Well, the answer digit should always go directly above the digit we’re dividing.
Since we’re dividing the digit 9, our 2 should go right above the 9.
Okay, now we multiply… 2 times 4 is 8, and the 8 goes below the 9 so that we can subtract to get our remainder.
9 minus 8 is 1, so our remainder is 1.
Now at this point in our basic one-step division problems, we would re-write our remainder up in our answer with a little ‘r’ next to it.
But we aren’t going to do that yet because this is long division and we still have more digits to divide (the ones we’ve been ignoring).
Since we’re going digit-by-digit, let’s stop ignoring the next digit in our dividend (the 3).
Now you might think that our next division step is to divide that 3 by the 4. But it’s not quite that simple.
We had a remainder from our last division step, and we can’t just forget about that.
We need to combine that remainder with our next digit and divide them both together.
We do that by bringing down a COPY of the next digit (the 3) and put it right beside the remainder (which is 1).
When we do that, it looks like our remainder is 13. It’s kind of like our remainder is teaming up with the next digit over.
And if you think about it, that makes sense because
the digits that we were ignoring during our first division step really are part of the remainder, because we still need to divide them.
Okay, so bringing down that next digit makes our remainder bigger.
And that’s good because before, the remainder was so small that 4 couldn’t divide into it.
But now it’s 13, and 4 will divide into 13.
So we ask, “How many ‘4’s will it take to make 13?”
Well, three ‘4’s would be 12, and that’s really close without being too big. So let’s put 3 in our answer line.
Yep - it goes right over the 3 because that is the next digit we were dividing in this digit-by-digit process.
And then 3 times 4 is 12 which we put right below the 13 so that we can subtract to get the next remainder which will also be 1.
See how we’re just repeating the basic division procedure? But we’re going further down the screen as we do.
Alright, now that we have a new remainder, it’s time for our next division step.
Let’s stop ignoring the last digit in the dividend (the 6) and bring down a copy of it to team up with our new remainder.
Together, they form a remainder of 16.
Ah ha! That’s good because it’s gonna be easy to divide 4 into 16 because 16 is a multiple of 4.
It takes exactly fours ‘4’s to make 16. So we put a 4 in the last place of our answer line,
and then we write the 16 below our new remainder.
Now if we subtract 16 from 16, we see that our last remainder will be zero, which means there’s no remainder left.
That’s great! We solved the whole division problem digit-by-digit by breaking it up into three basic division steps.
And now we know that 936 divided by 4 equals 234.
And we also know why they call it long division!!
In fact, that was so long, I think I need a coffee break…
Oh man…
that was some looooooooooooong division!
Wheew… let’s see…
Okay, so that problem had a three-digit dividend and it also had three division steps.
But the number of steps isn’t always the same as the number of digits we have.
And that’s because the number of steps also depends on how big our divisor is.
To see what I mean, let’s work two division problems side by side.
These both look like the basic one-step division problems that you did in the last video, don’t they?
But as you‘ll see, one of them is actually a two-step problem.
Let’s start with the first problem: 72 divided by 8.
We just ask, “How many ‘8’s does it take to make 72 (or almost 72)?”
Well that’s easy! On our multiplication table you can see that 72 is a multiple of 8.
8 × 9 = 72 So we put 9 in our answer line, and we write 72 below, and we see that we have no remainder.
Now let’s try the next problem: 72 divided by 3.
If we ask, “How many ‘3’s will it take to make 72 (or almost 72)?”, we can see that the answer is not on our multiplication chart.
The biggest multiple of three listed there is 30 which isn’t even close.
The reason is that this should really be a two-step problem.
Let’s try using the new digit-by-digit method we just learned.
Instead of asking, “How many ‘3’s make 72?”, let’s just focus on the first digit and ask,
“How many ‘3’s does it take to make 7?”
Ah - that’s easy. Two ‘3’s would give us 6, which is very close.
So let’s put a 2 in the answer line right above the 7.
Then we multiply 2 times 3, and that makes 6.
And we subtract 6 from 7 to get a remainder of 1.
Now for the second step…
We bring down a copy of the next digit (the 2) and we combine it with the 1 to get a new remainder of 12.
Then we ask, “How many ‘3’s does it take to make 12?”
and the answer to that is exactly 4.
So we write a 4 in the answer line, and 3 × 4 = 12.
12 − 12 = 0 So we have no remainder!
We’re done! 72 divided by 3 is 24.
Now here’s the interesting thing about these examples.
The first problem could have been a two-step problem also.
If we had taken it digit-by-digit, we would have first asked, “How many ‘8’s does it take to make 7 (or almost 7)?”
But the answer would have been zero since 8 is too big to divide into 7.
We would have put zero in our answer line and the remainder would have just been 7.
So basically, we just skipped that step. And that’ll happen with digit-by-digit division sometimes.
If the number is too small to divide into, you just put a zero in the answer line and you move on to the next digit.
Okay, now that you know the procedure for long division, are you ready to see a really long problem?
Good - I thought so!
Let’s divide 315,270 by 5.
Now don’t worry… it’s really not that hard if you just go digit-by-digit.
I’m gonna work the problem pretty fast, so don’t worry if you don’t follow all the math.
Just focus on the repeating division process as we go along. Are you ready?
The first digit is 3. How many times will 5 divide into 3? Zero.
5 is too big. So let’s just skip that step and combine our first digit with our next digit.
So how many times does 5 divide into 31? Six.
6 times 5 is 30. And 31 minus 30 gives us a remainder of 1.
Now on to the third digit… We bring a copy of it down to join with the remainder.
And we ask how many times will 5 divide into 15? Three.
3 times 5 is 15. And 15 minus 15 is zero.
On to the next digit…
Now even though our previous remainder was zero, we still bring down a copy of the next digit.
Now we ask how many times will 5 divide into 2? Zero.
5 is too big, so we need to move on to the next digit and bring a copy of it down also.
There, that’s better.
Now we ask, how many times will 5 divide into 27? Five.
5 times 5 is 25. And 27 minus 25 gives a remainder of 2.
…now for that last digit, which is a zero.
And you might wonder, “Why do we even have to bring a copy of a zero down? Isn’t that nothing?”
But the zero is an important place holder… and when we bring a copy of it down,
it changes our remainder of 2 into a remainder of 20. Now that’s a big difference!
Now we ask, how many times will 5 divide into 20? Four.
4 times 5 is 20. And 20 minus 20 is zero.
Yes! We’re done! There’s no more digits to divide.
And you can see that our final answer is: 63,054.
Alright… that’s the procedure for long division.
As you can see, it’s kinda complicated, so don’t get discouraged if you’re confused at first.
Like almost everything, it just takes practice.
So, as you get ready to practice some long division problems on your own, here’s a few tips that will help you out.
First: If you haven’t already done it, memorizing your multiplication table will really help with division.
Second: When you’re working problems, it’s really important to write neatly and stay organized.
If your writing is messy, it might be hard to keep your columns lined up and that could lead to mistakes.
And if that’s the case, try using graph paper to help keep things lined up.
Third: Start with some smaller two or three-digit dividends so you only have a few division steps to do.
Then work up to the longer problems.
And last of all: After each practice problem you do, check your answer with a calculator.
That will let you know right away if you’ve made any mistakes so you can correct them, and most importantly, learn from them.
And it will give you practice with a calculator, which is also important.
Alright, that’s all for this lesson.
Thanks for watching Math Antics, and I’ll see ya next time.
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