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
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
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.