Lesson Video: Dividing a Three-Digit Number by a One-Digit Number: Subtracting Partial Quotients Mathematics • 4th Grade

In this video, we will learn how to divide three-digit numbers by one-digit numbers using partial quotients.

17:49

Video Transcript

Dividing a Three-Digit Number by a One-Digit Number: Subtracting Partial Quotients

In this video, we’re going to learn how to divide three-digit numbers by single digits. And to do this, we’re going to use something called partial quotients.

Now, one of the methods that we use when we learn how to divide is repeated subtraction. Do you remember how this works? If we want to find the answer to 24 divided by three, for example, we can start at 24 and see how many threes fit into it. And we can do this by taking away three at a time or repeatedly subtracting. 24 take away three is 21. 21 take away three is 18. And we just keep subtracting threes until eventually we reach zero. And at that point, there are no more threes to take away. And because we subtracted three eight times to get to zero, we can say 24 divided by three equals eight. And this kind of repeated subtraction works quite well when we’re thinking about two-digit numbers.

But what if we have to divide a larger number? As we said at the start in this video, we’re gonna be thinking about dividing three-digit numbers by single digits. So what if you want to find the answer to 138 divided by three? It’s going to take us forever to keep subtracting threes, isn’t it? And we need a really long number line. There’s gotta be a quicker, more efficient method we could use. Well, there is, and it involves using what we call partial quotients, which sounds complicated but isn’t really. The word “quotient” just means the answer to a division calculation. It’s what we get when we divide one number by another. And the word “partial” is just talking about part of something. So what this really means is we’re going to be finding the answer to 138 divided by three a bit at a time.

We will divide 138 by three, but we’re going to do it a part at a time. Let’s show what we mean by this. Now, when we’re using this method, we can write the calculation 138 divided by three like this. It’s a way of writing divisions, but it’s interesting, isn’t it? There isn’t a division symbol in there at all. This is where we write the number we’re dividing. And this is where we write the number we’re dividing by or the divisor. So we could read it as, how many threes are in 138? Now as we’ve said already, we could repeatedly subtract threes, but this would take such a long time to get the answer zero.

Is there a larger multiple of three we could subtract to get to zero quicker? Well, we know that 10 threes are 30. What if we subtracted 10 lots of three at a time? In other words, what if we break up 138 into chunks that are worth 10 times three? Partial quotients, that would definitely be quicker than subtracting three every time, wouldn’t it? Let’s see where it takes us. Now 138 already has a three in the tens column. So if we take away 30 from this, it’s quite quick to do. The answer is 108. Now, as we use this method, it’s going to become more and more important to keep a record of what we’ve subtracted. So next to 30, we need to write down that this is the same as 10 lots of three.

We know definitely there are 10 lots of three in 138. And because we’ve got 108 left, we can see we can take away another 10 lots of three too. This time, we could count backwards in 10s three times from 108. So that’s 108 and then 98, 88, 78. 108 take away 30 is 78. And you know, we could keep going, keeping taking away these partial quotients, these chunks of 10 threes. 78 take away 30 will take us to 68, 58, 48. And we’ve got just enough left to take away one more lot of 10 threes. 48 take away 30 will leave us with 38, 28, 18.

Now, remember, this is all about getting down to zero. And we’ve only got 18 left. And we should know a times tables fact that will help us subtract all of that 18 in one go. Six threes are 18, aren’t they? So instead of making lots of smaller jumps of three, we’ve made one, two, three, four jumps of 10 times three and then a jump of six times three. So how many threes was that altogether? In other words, what’s 10 plus 10 plus 10 plus 10 plus six? Well, four 10s are worth 40. And if we add six, this gives us 46. And so we can say 138 divided by three is equal to 46. And just to show you often with this type of division, we write the answer on the top like this, 46.

Now, that was fairly efficient, wasn’t it? We found the answer quite quickly. We only needed to subtract five times, which is a lot less than we would have done if we’d have had to take away threes. But, you know, we could even work even more efficiently. And the way to do this is to give it some thought beforehand, to look at the calculation, how many threes are in 138, and to ask ourselves the question, “What’s the largest multiple of three I know that’s less than 138?” In other words, how close can I get to 138? I could take away 30 each time. But are there some larger multiples than I know?

This is where having a pen and paper can help because we can jot down some facts before we start. Because if I know that 10 threes are 30, I could double this and know that 20 threes are worth 60, so I could take away 20 lots of three at a time. But what if we double the fact again? We can see that 40 threes must be worth double 60 or 120. And that’s really close to 138. Let’s see how quickly we can find the answer by subtracting 40 lots of three first.

We still have eight in the ones place. Three 10s take two 10s leaves us with one 10. And we don’t have any 100s because we’re taking away 100 from 100. So we found a partial quotient of 40 lots of three. And we’ve only got 18 left. And just like before, we know that there are six lots of three in 18. So if we take away six lots of three, we’ll be left with zero. This time, we got to the answer in two steps. We aren’t going to get much better than that, are we? First, we subtracted 40 lots of three and then six lots of three, and 40 plus six equals 46. We got the same answer, but a lot more quickly. And the reason we were able to get there so quickly is because we took a few moments before we started and asked ourselves, “What is the largest multiple of three I know that’s less than 138?” The bigger the multiple, the less steps you’re gonna have to do.

Now in order to get better at using this method and also to really understand how it works, it makes sense for us to practice a few questions. We’re going to go through three questions now where we have to divide three-digit numbers by a single digit. And each time, we’re going to use this partial quotient method. We’ll take it quite slowly to begin with, because it’s quite new to us. But hopefully, by the end, we’ll be working a lot more quickly. And you’ll be able to see just how useful this method can be. So let’s start with question one.

Benjamin is calculating 450 divided by five using the partial quotient method. Help him to find the quotient from his calculation.

In this question, we’re told that Benjamin is trying to work out the answer to a division question. We know this because we’re given it in the first sentence. We’re told that he’s calculating 450 divided by five. But you know, we can also see the calculation written underneath here. It might not have a division symbol, but wherever we see numbers written like this or sometimes like this, we know we need to divide to find the answer. We need to find how many fives there are in 450. Now, like with any calculation, there’s more than one way Ben could find the answer. But we’re told that he’s using the partial quotient method.

Now this method is easier to understand that it looks. The word “partial” just means part of something. And a quotient is simply the answer to a division. In other words, Benjamin is finding out how many fives there are in 450 a part of the answer at a time. He’s breaking 450 into chunks. And we’re told that we need to help him find the quotient from his calculation. Now, one way to find out how many fives are in 450 would be to keep subtracting fives. 450 take away five leaves us with 445. And if we take away another lot of five, that’s 440. It’s gonna take us quite a while to get all the way down to zero, isn’t it?

So when we use the partial quotient method, we ask ourselves a question. What’s the largest multiple that I can think of that I could subtract from this number? Instead of taking away one lot of five each time, can you think of something else we could subtract? Well, we know that 10 fives are 50, so we could take away 10 lots of five at a time, take away 50s. But we could even go one better than this. What if we double the fact? If 10 fives are 50, 20 fives are worth double 50 or 100. If we take away 100, we’re going to get to the answer much quicker, aren’t we? So let’s try subtracting this partial quotient. And it might help us to consider what Ben’s thinking about at each step.

So to begin with, he’s going to look at his number 450 and think to himself, “Well, I know there’s definitely 20 lots of five in 450, so I’m going to take away 20 lots of five or 100.” And of course, 450 take away 100 leaves him with 350. Now he can look at 350 and think to himself, “Well, I can take away another lot of 25s.” 350 take away 100 is going to give him 250. And he’s still got enough to keep going. 250 take away 100 leaves him with 150. 150 take away 100 leaves him with 50. And now he can’t take away anymore chunks of 25s, can he? He’s only got 50 left. But Benjamin knows how many fives there are in 50. And if he takes away 10 lots of five, he’s going to arrive back at zero.

So by subtracting a part at a time, Benjamin split up the answer to this division calculation or the quotient into bits. He knows there are one, two, three, four lots of 25s and one lot of 10 fives. Now, how many fives is that altogether? Well, we know that four lots of 20 are worth 80. And if we add 10, we get the answer 90.

We’ve subtracted multiples of five to find the answer. This is called the partial quotient method. And we found we had to take away 90 lots of five to get back to zero. And that’s how we know 450 divided by five equals 90.

Evaluate 848 divided by four using the partial quotient method.

In this question, we’re asked to evaluate or find the value of 848 divided by four. Or in other words, how many fours are there in 848? Now whilst we could use times tables facts we already know if we were dividing a two-digit number by four or even use repeated subtraction by taking away four each time, we need to use a method that’s a little quicker when we’re dividing a three-digit number like this by four. And that’s why we’re told to use the partial quotient method. And this involves breaking up the number 848 into smaller parts. We might not know how many fours there are in 848 straightaway, but we do know how many there are in part of 848. Then we can look at what’s left and look at another part and so on.

So before we start, let’s think about some multiples of four that we could take away. We don’t want to be subtracting four and then four and then four and then four, do we? Now, we know that 10 fours are worth 40. Now that could come in useful. But 848 is still quite a large number to be taking away 40 from each time. What if we multiply this fact by 10? What are 100 fours? Well, this is quite a straightforward fact too. 100 fours are 400. What if we start off by taking away 100 fours at a time? That will take us back to zero quite quickly, won’t it? So we’ll start off by subtracting a chunk of fours, 100 lots of four or 400.

Now our number 848 has eight 100s in it. And if we take away four of those 100s, we’re gonna be left with 448. Now we can see that we’ve got just over 400 left, haven’t we? So there must be another 100 fours we could subtract. 448 take away 400 leaves us with just the tens and the ones, 48. Now, what could we subtract from 48? Well, if we use facts we already know, we could take away all 48 at once. We know 10 fours are 40. That means 11 fours must be 44 and 12 fours must be worth 48. So if we take away those 12 fours, it’ll bring us back to zero. So how many fours did we find there were in 848? There were 100, another 100, and 12 more. And 100 plus 100 plus 12 equals 212.

We found the answer to 848 divided by four using the partial quotient method. The number of fours there are in 848 is 212.

Evaluate 216 divided by eight. Hint: use partial quotients to help you.

In this question, we need to find the answer to 216 divided by eight. And we’re given a hint. We’re told that we can use partial quotients to help us. In other words, instead of thinking about how many eights there are in 216 eight at a time, we can split the number 216 into parts. And we can find the answer one chunk at a time. So to begin with, let’s write out our calculation. And the way we do this is by putting down the divisor. That’s the number we’re dividing by first. Then we draw a series of lines. It will always look exactly like this. This is one way to draw it. And then underneath the top line, we write the large number that we’re dividing.

So this calculation asks us, “How many eights are there in 216?” Now what multiples of eight could we take away from 216 to take us down to zero quickly? We could subtract eight, then eight, then eight, then eight, and so on, but that would take a really long time. A fact that we could use to help us is 10 eights. 10 eights are worth 80. But you know, we could get even closer to 216 if we double this fact. If we know 10 eights are worth 80, then 28 must be worth double 80 or 160. Now we can definitely see that there are 20 eights in 216. So let’s take away 160 straightaway.

Now 216 take away 100 leaves us with 116. And if we take away 60, this takes us back to 56. And so we can say 216 take away 160 equals 56. Now we know how many eights are in 56, don’t we? This is a times table that we’ve already learned. There are seven eights in 56. And so if we take away our last seven eights, we’re going to be left with nothing.

We’ve used partial quotients to help us find how many eights there are in 216. And we took away 20 lots of eight and then another seven lots of eight. And that’s how we know 216 divided by eight equals 27.

So what have we learned in this video? We’ve learned how to divide a three-digit number by a single digit using partial quotients to help.

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