### Video Transcript

In this video, we’ll learn how to
share quantities into two- and three-part ratios. By this stage, you should feel
fairly confident in identifying and using ratio notation as a way of comparing
amounts. You should also know that
simplifying a ratio is much like working with fractions. We look for common factors and
divide through until the numbers are coprime. That is, they share no factors
other than one.

Now, we’re going to look to extend
these ideas and see how we can share given amounts into ratios with two or more
parts. And we’ll consider two equally
valid methods. Before we get into the
nitty-gritty, let’s look at how we could solve a question pictorially.

Bart and Lisa share 20 sweets in
the ratio two to three. How many sweets does each child
get?

We’ll begin by recalling what we
actually mean by this ratio two to three. This means that for every two
sweets Bart gets, Lisa will get three. It follows that we expect that if
we share these sweets out in this manner, Lisa will get more sweets than Bart. So let’s begin by giving Bart two
sweets, and then we’ll give Lisa three. There are still quite a few sweets
left. So let’s do that again. We give Bart two further sweets,
and we give Lisa three.

Notice at this stage Bart has four
sweets and Lisa has six. If we put these in a ratio that is
four to six, we can simplify this back down to our original ratio two to three. That’s a really good way to check
our answer. But there are still more sweets
left. Let’s keep going until we’ve used
them all up. Bart gets two more, and Lisa gets
three. Bart gets another two, and Lisa
gets the three that are left.

So we’ve used all of these sweets
up. And for every two sweets we’ve
given Bart, we’ve given Lisa three. We now see that Bart has eight
sweets and Lisa has 12. Well, that’s all fine and well. But we really don’t want to do this
every time. So we’re going to need to find a
quicker method.

This method involves thinking about
the ratio in terms of its individual parts. We go back to our original
ratio. It consists of two parts here and
three parts here. Two plus three is equal to
five. So our ratio consists of a total of
five parts. Let’s draw a bar model to represent
this. Our bar model needs to have five
parts, just like our ratio. And, of course, we need to fill
this bar with our 20 sweets. So how do we share 20 sweets into
five parts or five boxes? What mathematical symbol means
share?

Well, the symbol divide means to
share. So we divide 20 by five, and that
gives us four. In doing this, we’re working out
the value of one part. One part is worth four sweets. And now we know this, we go back to
the start and remind ourselves how many parts each child gets. Well, Bart gets two parts — I’ve
shaded his parts yellow — whereas Lisa gets three parts — I’ve shaded that in
pink.

We now know that each individual
box contains four sweets. One part is worth four. And so to work out the number of
sweets Bart gets, we multiply four by two. That gives us eight. Similarly, we can multiply four by
three to work out the number of sweets Lisa gets; that’s 12. And a really quick way we can
double check whether we’ve performed the correct calculation is to check that our
final values add up to the original. Well, eight plus 12 is indeed 20 as
required. So we’re done.

And hopefully, we see that this is
a much more efficient method than repeatedly giving two sweets and three sweets and
so on, especially when it comes to sharing much larger amounts. Note that these processes can be
used with integers as we have done — that’s whole numbers — but also with noninteger
numbers.

In general, the steps are as
follows. We begin by adding the numbers we
have in our ratio. This tells us the number of parts
that we’re sharing into. So we divide our original amount by
this value. That then tells us what one part is
worth. So we take that value and we
multiply each bit of our ratio by that number. This is sometimes called the
add-divide-multiply method for fairly obvious reasons.

It’s also important to notice that
when we’re not actually sharing an amount into a ratio, we use similar sets,
although our first step won’t necessarily be adding the numbers. Let’s look at an example of how
this works.

Divide 48 kilograms in the ratio
two to one.

One way we have to answer this
question is sometimes called the add-divide-multiply method. Let’s recall the steps that we
take. The first thing we do is establish
how many parts we have by adding the numbers in our ratio. Well, we’re sharing in a ratio of
two to one. So we add two and one to give us
three. That tells us that we’re dividing
our original amount into three parts.

Next, we do that division. We divide the original amount by
this total. In this case, that’s 48 divided by
three, which is equal to 16. That tells us what one part is
worth. So here we see that one part is
worth 16 kilograms.

Finally, we take this number and we
multiply it by each bit in our ratio. Well, the first bit of our ratio is
two. Two times 16 is 32. So when we share 48 kilograms in
the ratio of two to one, our first bit is 32 kilograms.

Now, of course, the second bit is
simply one part. So we don’t really need to do this
multiplication. But we will for completion. One times 16 is 16, remembering of
course that order matters. So we need to list the largest
value first. We see that when we share 48
kilograms in the ratio two to one, we get 32 kilograms and 16 kilograms.

Now, using a step-by-step method is
all fine and well. But it’s really important to
understand what we’re doing at each step. Let’s look at an example where we
might be inclined to blindly follow these steps without really thinking about what
we’re doing.

Chloe and Michael shared their
lottery winning in the ratio of three to two. Michael received 360 pounds. How much did Chloe receive?

When we answer this question, we’re
going to need to be really careful. We’re told that Chloe and Michael
share their lottery winning. And at first glance, we might see
the ratio of three to two in the value of 360 pounds and think we’re going to share
this value in the ratio three to two. In fact, this is the amount that
Michael received. So in essence, this has already
been shared for us. And our job is then to work out how
much Chloe receives.

So let’s think about the steps we
take when we’re sharing into a ratio and what they actually mean. The first thing we do is add the
numbers in our ratio. This tells us the total number of
parts we have. Then we divide our amount by this
number to find the value of one part. Once we know the value of one part,
we can go on to the next step. We multiply each bit of our ratio
by this number. And so, in this case, we can
disregard the first step.

But what we are going to do is
divide to find the value of one part. Chloe and Michael share their
winning in the ratio of three to two. So the second part of this ratio
represents the number of parts that Michael gets. He gets two parts. So 360 pounds must be worth two
parts. We’re going to divide this number
by two to find the value of one part. 360 divided by two is 180. So we see that one part is worth
180 pounds.

And so, once we’ve done this,
what’s our final step? Well, we’re going to multiply 180
pounds by the number of parts that Chloe receives. Chloe receives three parts, so we
multiply 180 by three. This gives us a value of 540
pounds. Chloe receives 540 pounds. And, in fact, we could check our
answer.

We’ve said that Chloe receives 540
pounds and Michael 360. So let’s put this in a ratio and
simplify it. We could do this in a few
steps. But in fact, both of these numbers
are divisible by 180. Since the units are the same, we no
longer need them. And we see that to simplify 540 to
360, we get three to two. This is the same as the ratio given
to us in our question. So we know we’ve done it
correctly. Chloe gets 540 pounds.

In our next example, we’ll consider
a slightly different method for sharing a number into a given ratio.

David and Isabella share 30
cupcakes in the ratio two to three. How many do they each get?

One way we have to answer this
question is the add-divide-multiply method. But there is another method which
is equally valid. And that involves thinking about
bar modeling and fractions. The first step is the same as the
add-divide-multiply method. We look at our ratio. That’s the ratio two to three. We then add the numbers in our
ratio. Two plus three is equal to five,
meaning we’re looking at a total of five parts in this question. And so a bar model will be made up
of five parts, as shown.

Remember, order is important. So David gets two of these
parts. That’s represented by the yellow
bits. And Isabella gets three. That’s represented by the pink
parts. Now, we can think about this in
terms of fractions. And we can see that the portion of
the bar that I’ve shaded yellow represents two-fifths of the whole, whereas the
portion that I’ve shaded pink represents three-fifths of the total.

We can extend this into the number
of cupcakes and say that, well, if there are 30 cupcakes, David gets two-fifths of
these. Similarly, Isabella will get
three-fifths of these. By recalling that “of” can quite
regularly be interchanged with the multiplication symbol, we see that we can answer
this question either by finding two-fifths of 30 and three-fifths of 30 or by
multiplying two-fifths by 30 and three-fifths by 30.

Let’s multiply two-fifths by
30. We’ll write 30 as a fraction. It’s 30 over one. Next, we cross cancel. We divide the denominator of our
first fraction by five and the numerator of our second by five. Two times six gives us 12, and one
times one gives us one. But 12 ones is simply 12. And so we see that David gets 12 of
the cupcakes.

To find the number that Isabella
gets, we could subtract this from 30. But let’s multiply again. We do three-fifths times 30 over
one and, once again, divide through by five. This gives us three times six,
which is 18 over one, or simply 18. David gets 12 cupcakes, and
Isabella gets 18.

Note that, of course, we can check
our answer by checking that 12 and 18 do add up to 30, which they do. Note also that at this stage, we
could have found the value of one-fifth by dividing 30 by five. Then whatever value we get, we
double to find the value of two-fifths. Either method is perfectly
valid.

In our next example, we’ll go back
to the add-divide-multiply method and consider how it will work when sharing using a
three-part ratio.

Divide 72 centimeters in the ratio
two to three to one.

Let’s recall one of the methods we
have for sharing into a ratio. It’s called the add-divide-multiply
method. In this method, the first thing we
do is we add the numbers in our ratio. This will tell us the total number
of parts we’re sharing into. Our ratio is two to three to
one. And two plus three plus one is
six. So we have a total of six
parts.

Step two is to divide the amount
we’ve been given by this number. This will tell us what one part is
worth. We’re dividing 72 then by six, and
that gives us 12. Of course, we’re actually working
in centimeters. So we can say that one part is
worth 12 centimeters.

Our final step is to multiply each
bit of the ratio by this number. Essentially, if we know the value
of one part, we can find the value of two by timesing it by two, we can find the
value of three by timesing it by three, and so on. Two multiplied by 12 is 24, so the
first bit we get is 24 centimeters. Three times 12 is 36, so our second
bit is 36 centimeters. Then one multiplied by 12, which we
don’t really need to do but we’ll do for completion, is 12. And the third bit is 12
centimeters.

When we divide 72 centimeters in
the ratio two to three to one, we get 24 centimeters, 36 centimeters, and 12
centimeters. In fact, 24 plus 36 plus 12 gives
us the original 72 as we required. So we know that we’ve probably done
this correctly.

In our final example, we’ll go back
to the fraction method and consider how it works when using a three-part ratio.

Three friends win 6000 pounds. They decide to share the money in
the ratio three to five to four. Calculate the amount each one
receives.

To answer this question, we’re
going to use a fractional method. By considering the ratio three to
five to four, we want to work out what fraction of the total amount each friend
wins. Now, for ease, we’re going to name
these friends 𝐴, 𝐵, and 𝐶. And the first thing we’re going to
do is add the numbers in our ratio. What this does is tells us the
total number of parts we’re sharing the winnings into. Three plus five plus four is
12. And so we’re sharing this 6000
pounds into 12 parts.

Remember, order matters. So 𝐴 gets three of these parts, 𝐵
gets five, and 𝐶 gets four. In terms of fractions then, we can
say that if there are a total of 12 parts and 𝐴 gets three of these, 𝐴 must get
three twelfths of the winning amount. In fact, we can simplify three
twelfths to a quarter. So 𝐴 gets one-quarter of 6000
pounds. In a similar way, since 𝐵 gets
five of the parts, we know 𝐵 gets five twelfths of the money. Finally, 𝐶 gets four parts. So 𝐶 gets four twelfths of the
money. Four twelfths simplifies to a
third. So 𝐶 gets one-third of 6000
pounds.

We’ll work out then how much 𝐴
gets by finding a quarter of 6000. That’s 6000 divided by four, which
is 1500. We’ll come back to 𝐵 in a moment
because it requires an extra step. But to find a third of 6000, we
divide 6000 by three; that’s 2000. We’ll find for 𝐵 one twelfth of
6000 first. That’s 6000 divided by 12, which is
500. So one twelfth is 500 pounds. Five twelfths will be five lots of
500 pounds. That’s five times 500, which is
2500 pounds. The three friends receive 1500
pounds, 2500 pounds, and 2000 pounds, respectively.

In this video, we’ve seen that when
we share in a ratio, we’re really interested in finding the value of one part of
that ratio. We saw that there are a couple of
methods that we can use. One of these is the
add-divide-multiply method. We add the numbers in our ratio
together, divide the original amount by this number, and that tells us what one part
is worth. When we know what one part is
worth, we multiply each bit of our original ratio by this number to share the
amount.

But we also saw that sometimes it
can be efficient to use fractions to calculate this. We saw that we can check our
answers by either adding the final numbers we get and checking that this gives us
the original amounts or by putting them into a ratio and simplifying them back. When we do this, we should get back
to the original ratio.

Finally, we saw that very
occasionally we won’t actually be given the entire amount that we’re sharing
into. Instead, we’ll be given what one
person receives. And very occasionally, we’ll be
given the difference between what they receive. And so we should always make sure
that we are being asked to share the amount given before following these steps.