www.mathcentre.ac.uk/.../Ratios.mp4

Today's session is on ratio.

I'm going to explain what a ratio is

and how ratios are used

in different situations.

So to start off with what is a ratio?

Well, a ratio is a way of comparing

amounts of ingredients.

Ratios can be used to compare

weights, money, length and so on.

So if we take this example

we've got a model boat

whose length is 1 metre

and the real boat

whose length is 25 metres.

Then we say the ratio of the

length of the model boat to the real boat

is 1 to 25.

Notice we've just used the numbers

without the unit (metres)

and we've used the colon

to represent the ratio.

Ratios are used to describe quantities

of ingredients in mixtures.

For example, in the
pharmaceutical trade

when you're making medicines,

or in the building trade

when you are making cement or mortar,

or at home when you're making up food

you use different quantities
in different proportions

and if you don't get them right

then things go wrong.

So it's very important to know

what quantities you've got and
in what ratio.

So for example, if we have

mortar for building brick walls.

Mortar is made up by mixing

two parts of cement to
seven parts of gravel by volume

and we write that ratio as 2 to 7.

Again notice we've used

the numbers without the units

and the colon to represent the ratio.

When we're making pastry at home,

when we're making pies and tarts,

we mix four ounces of flour

with two ounces of margarine

And that ratio would be 4 to 2.

But in this case,

if you look at the numbers,

they've got a factor of two in common

So we can simplify ratios just in the same

way as we simplify fractions.

We can divide by the common factor,

so we divide 4 by 2

and 2 by 2 to give 1.

So 2 to 1 is the simplest form
of the ratio 4 to 2.

But both of the ratios are equivalent,

because the relationship of the numbers
involved stays the same.

If we take this example

250 to 150

We can simplify this ratio.

We divide both by 10

to get 25 to 15

And then we can divide both by 5

5 into 25 will give me 5

5 into 15 will give me 3

We can't divide anymore,

so this is the simplest form.

5 to 3 the simplest form of

the ratio 250 to 150.

But all three ratios are equivalent

because the relationship of the numbers

is exactly the same.

In the same way, we can actually

simplify this ratio: 1 to 1.5

In ratios we like to have whole numbers

and in this ratio you can see

that we have a decimal.

To get rid of the decimal

we can multiply both sides of the

ratio by 10

and we still have an equivalent ratio.

Because, again, the relationship

between the numbers is the same.

So we multiply the 1 by 10

you get 10

Multiply 1.5 by 10

you get 15

10 to 15 we can simplify that.

Divide both sides by 5.

5 into 10 gives me 2

5 into 15 will give me 3

2 to 3 is the simplest form

of the ratio 1 to 1.5

Similarly, when we have fractions

If we had this ratio:

a quarter to five-eighths,

it just doesn't look right.

We want to express that ratio

in terms of whole numbers

in its simplest form.

So what we do first is we write
both as fractions over 8

in terms of eighths.

A quarter is two eighths

and now the ratio is two eighths to five eighths.

And now it's dead simple

All we have to say is that is 2 to 5.

We multiply both ratios by 8

And 2 to 5 is the simplest ratio

for the ratio a quarter to five-eighths.

But again all three ratios are

equivalent because the relationship

between the numbers is exactly the same.

Moving on,

we must have the numbers
in the ratios

having the same units.

So if we have this ratio

15 pence to 3 pounds,

we cannot say that the ratio is 15 to 3

and then simplify that to 5 to 1

Because we didn't start off with

the numbers having the same units

the relationship between
the numbers is not the same,

because as I say,

we didn't start off with these numbers

having the same units.

So we must convert the numbers

to the same units

and we choose whichever unit is
appropriate

In this case, it's obvious we must

change them to pence.

So we say the ratio is 15 to 300

as there's 300 pence for 3 pounds

and then we simplify as normal.

We divide both sides by five.

5 into 15 is 3

5 into 300 is 60

And then we can divide by 3

3 into 3 is 1

3 into 60 is 20

And notice these two ratios are

not the same, they're vastly different.

They're not equivalent because

the relationship between the numbers

is not the same.

So it's very important in ratios

that you start with numbers

that have the same units.

If they're not,

then you convert them to the same units

and then simplify if appropriate.

As I said before,

ratios are extremely useful

in lots of different circumstances.

They can be used to divide and

share amounts of different quantities

like money, weights, and so on.

So if I take this problem

just say I had an inheritance of £64,000

and it was to be shared between two people

Mrs Sharp and Mr West

in the ratio 5 to 3

What I want you to do is work out

what each one of those gets.

And that's a lot of information to take in

so what I do first is

I start off with a diagram

I've got the total inheritance of £64,000

and I divide it

between Mrs Sharp

and Mr West

in the ratio 5 to 3

And we want to work out

what each gets.

What we do first is we work out

the total number of parts that

their inheritance is split up into.

Well, we use the ratio for that.

It's five parts for Mrs Sharp

and three parts for Mr West

so altogether that is eight parts

Then we work out what the total value

of one part of the inheritance would be.

Now we know that the total inheritance

is £64,000

so one part

equals 64,000 divided by 8

and that is £8000

And then the rest is easy.

All we have to do now is
take Mrs Sharp

and she has five parts

and that is 5 multiplied by £8000

which works out to be £40,000

And then Mr West

he has three parts

and that is 3 multiplied by £8000

which is £24,000

An awful lot of money!

But what if I made a mistake?

How can I check my two answers?

How can I check that Mrs Sharp did get
£40,000

and Mr West got £24,000?

Well a very simple check

is to add up these two values

and if they add together

to make up the total inheritance

then we think we've done our
calculations properly.

So a quick check:

£40,000 plus 24,000 does equal £64,000

For a complete check though

we can take the two amounts

and see that they will actually make an
equivalent ratio

to the ratio that we started off with 5:3

So if we take our 40,000 that
Mrs Sharp got

and then the 24,000 that Mr West got

and cancel them down,

we cancel by 1000

then we cancel by 4

so that would make 10 to 6

and then cancel by 2

so that will make 5 to 3

We do actually get the same ratio

that we started off with.

We're going to do another example.

It's an example which involves another

mixture: making concrete.

And with this, concrete is made by mixing

gravel, sand and cement

in the ratio 3 to 2 to 1

and in this problem we
start with concrete.

The amount of concrete
that we are going to make

will be 12 cubic metres.

And what I want to work out

is how much gravel will be needed

to make 12 cubic metres of concrete.

So we start with drawing a diagram

and that represents the concrete

and we know we want to make

12 cubic metres of concrete

and we know it's mixed

by mixing gravel, sand and cement

in the ratio 3 to 2 to 1

And we want to work out

the amount of concrete for 12 cubic metres

Well, first of all,

we work out the total number of parts

our concrete is divided up into

and we use our ratio for that.

It's 3 + 2 + 1 and that equals 6 parts

Now our concrete is divided up

into six parts

So one part must equal

our 12 cubic metres divided by 6

so that's 12 divided by 6 cubic metres

which works out to be 2 cubic metres.

Now we want to work out

how much gravel is needed.

Gravel is represented by 3 parts

so gravel, the amount that we want

equals 3 times 2 cubic metres

which is 6 cubic metres

and that's our answer.

But it's always good to check

and so we try and do the calculation

in a different way

and the way that I'd like to do it

is using fractions.

If we go back to the original diagram

we know that gravel is represented
by 3 parts

and the total is 6

so gravel is a half of
the total volume

and a half of 12 cubic metres is

6 cubic metres

so our answer is right

we've done a check.

But what if we did a similar problem

and we want to start off

with mixing our concrete

using gravel, sand, and cement

but we don't know the final volume of
the concrete

but we do know that we are given

6 cubic metres of sand

and an unlimited supply of
gravel and cement.

How much concrete can we make then

if we've got 6 cubic metres of sand?

Alright, we'll start the
question or the problem

with a diagram.

We know that the mixture is
still the same.

We use the same ratio

gravel to sand to cement

as 3 to 2 to 1

And we know that

we have 6 cubic metres of sand

but we want to work out

how much concrete we can make

with that amount of sand

and unlimited amounts of the other two.

Well, the number of parts that

the concrete is divided up into is still 6

But we know that 2 parts

is 6 cubic metres

because that's what we're given

so 2 parts equals 6 cubic metres.

So 1 part

equals 6 divided by 2

which is 3 cubic metres

Now the total number of parts of

the concrete is divided up into is 6

So the amount of concrete that is produced

is 6 times 3 cubic metres

and that is 18 cubic metres

Again, it's good to check our answer

and we'll do it in a different way

and we'll use fractions again this time.

We look at what we were given.

Sand is represented by 2 parts

and we know it has a volume
of 6 cubic metres.

Altogether, there are 6 parts
for our concrete.

So the fraction that represents sand

is 2 over 6, which is a third.

So a third of the total amount is
6 cubic metres

So the whole amount of concrete must be

3 times 6 cubic metres

which is 18 cubic metres

Here's another ratio problem involved
with ingredients

but this time the ingredients are to make

the Greek food houmous.

It's usually given as a starter

and there are four ingredients:

two cloves of garlic

are combined with

four ounces of chickpeas

and four tablespoonfuls
of olive oil.

I sound a little bit like
Delia Smith at this point

and the final secret ingredient is

the 5 fluid ounces of tahini paste.

Now when you combine these ingredients

together that's enough for six people

But what if I want to make houmous

for nine people?

What amounts do I have of these four
ingredients

to make it for nine people?

Well, we start off with what we've got

and what we know

We've got 2 cloves of garlic

with 4 ounces of chickpeas

4 tablespoonsful of olive oil

and 5 fluid ounces of tahini paste

and that makes enough for six people

What I do next is that I work out

what each of those ingredients

would be for one person.

So I have to divide

each of those numbers by 6

So that's 2 over 6

4 over 6

4 over 6

and 5 over 6

and then we cancel down if we can

In this case we can

that's one third.

Cancel four sixths to two thirds.

And this will be the same.

And the last one just remains the same:

five sixths

And now it's dead easy to work out

what amounts we need for nine people.

All we have to do is multiply by 9

So that's 1/3 multiplied by 9

2/3 multiplied by 9

and another 2/3 multiplied by 9

and then 5/6 multiplied by 9

And we work out these
calculations and simplify

3 into 9 is 3

3 into 9 is 3

and then 2 threes are 6.

and this works out to be the same

which is 6 because it's the same
calculation

3 into 6 is 2

3 into 9 is 3

5 threes are 15 over 2

which works out to be 7 and a half

So our final answer

for the ingredients

is 3 cloves of garlic

6 ounces of chickpeas

combined with 6 tablespoonfuls
of olive oil

and 7 and a half fluid ounces

of tahini paste

And that makes enough
houmous for nine people.

In a similar way,

you can use this method in conversion
problems

If we had the conversion that

1 pound is the same as 1.65 euros

and I wanted to work out

what 50 euros would be in pence

to the nearest pence

What I like doing first is to work out

what 1 euro is in terms of pence

So I start with

1.65 euros equals 100 pence

One euro would then equal

100 divided by the 1.65

And then to work out

what the 50 euros would be

I multiply this by 50

as 100 over 1.65 multiplied by 50

And that is 5000

divided by the 1.65

Now I am not going to do this by
long division.

I'll use my calculator

and I just type in the relevant numbers

5000 divided by 1.65

equals

3030 point 3 0 point 3 0 repeating

So 50 euros equals 3030 pence

to the nearest pence.

Which is 30 pounds and 30p

Well, that's the session finished
now on ratio.

Before I finish finally,

what I'd like to do is just remind you

of a few key points about ratio.

First of all, what is a ratio?

Well a ratio is a way of comparing

quantities of a similar type

When you write a ratio down

you use whole numbers

separated by colon.

The numbers should be in the
same units.

If they're not, you convert them

to the same units

by using one or the other of the
units involved

Just use your nous basically.

And then you simplify as appropriate.

In calculations involved in ratio

it is useful to work out the total
number of parts

the quantity is divided up into

and then work out one part represents.