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.
