www.mathcentre.ac.uk/.../Percentages.mp4

In this tutorial, we're going
to look at the meaning of percentages

and then we're going to do
calculations involving percentages.

And before we finish will look at

how to use the percentage
button on your calculator.

Now, percentages are very
familiar to us.

We see them in the media an
awful lot,

whether they're talking about the

percentage of A to Cs
gained by youngsters in exams,

or whether they're talking about the
success of the latest

medical procedures.

In addition, any of those of you who work
you'll pay income tax

and income tax is based on a percentage
of your salary.

The National Insurance
contribution is

a percentage of your salary.

In addition to that, when we go
shopping, most of the goods that

we buy have got VAT, Value Added Tax,
added on at a rate of 17 1/2%

So percentages are all
around us. They're part of our life.

So what does percentage
actually mean?

Well in mathematics our symbol that we
use is the percent sign.

And it means
out of 100.

Or as a mathematical process
'out of' means divide by 100

So let's have a look at an
example.

Let's say you get 85 percent in a test.

That means you got 85 marks out of
a possible 100 marks

Let's look at some
familiar percentages

75%
that means 75 out of 100.

Or if we write the fraction in
its lowest form, it's 3/4

Or if we write it
as a decimal 0.75

50%
That's 50 out of 100.

And its lowest form. That's
exactly the same as a half

and as a decimal 0.5

25% is 25 out of 100

In its lowest form, that's a quarter.

or 0.25 as a decimal

Let's look at 10%. That's a very
common one.

That's 10 out of 100

or 1/10 as a fraction in its lowest form

or 0.1 as a decimal

And 5%, that's just five out of 100.

And in its lowest form, that's 1/20

and 0.05 as a decimal.

Now it's worth noting here that

to find 50% an easy way of doing it

is just to find half of the number

or divide by two.

And to find 10% another common one

You just divide by 10,

finding a tenth of the number.

Let's look at writing a fraction
as a percentage.

Let's say we get 18 marks out of 20 in a test

and we want to write that as a percentage.

First of all, we write it as a fraction:

18 out of 20

Now a percentage is out of 100,
so we want this number at the bottom

the denominator to be 100.

Well 20 multiplied by 5 gives us 100

So in this case

we can just multiply 18 by 5.

Which gives us 90 out of 100.

So 18 out of 20 is the same as 90%

Now that's fine when it's an
easy number that we've got our

marks out of 20, I could easily
change into 100. I know to multiply by 5

But what happens if it's not quite
such a nice number

Well, let's have a look.

Let's take the example this time

Let's say we have 53 marks.

And this time let's have it out of 68.

Well, we write it as a fraction.
Again, 53 out of 68.

And what we're going to do

here we multiplied by 5 and

we multiplied the denominator by 5.

So both numerator and denominator were
multiplied by 5

Well, this time we're
going to multiply both by 100

Now this 100 here gives us
our percentage because

percentage is divide by 100.

So we're actually going to
calculate 53 divided by 68

multiply it by 100.

And that will give us our percentage.

And if we carry out that calculation,

we find that it comes to 77.94%

That's to two decimal places.

Or if we put it to the nearest percent
78%

Let's look at an example now where we're

finding the percentage of a quantity,

Now, in many countries it's customary
to leave a 10% tip when you have a meal

So let's say our meal costs £25.40

and we want to leave a 10% tip

Now we want 10% of £25.40

So in mathematical language

10% is 10 divided by 100

'of' is multiply

£25.40

And that equals £2.54,

so that's the tip that we would leave

Now 10% I said earlier that was an easy

one to calculate because you just

divide by 10.

But the method that I've shown here

is what you would use for

more complex percentages.

Let's have a look at another one.

Let's say we have a salesman who earns
2% commission

on orders that he gets in one month.

And let's say he has orders to
the value of £250,000

How much is he going to earn?

Well, we want 2% of £250,000

Again turn to our mathematical language

2% is 2 divided by 100

'of' (multiply) £250,000

Now in this one we can do some cancelling

because 100 goes into £250,000

So we've got twice 2500

Giving us an answer of £5000.

Now VAT, Value Added Tax, is
17 1/2% at the moment

So let's do some calculations where we

need to add on VAT

Let's say a computer costs

£634 plus VAT

Now we can look at this in
the same way as we did these examples

We can find 17 and a half percent of £634

That's the VAT and
we can add it on.

Well, let's do that.

So 17 1/2% of £634 is 17.5

I'm going to write it as .5

when I now turn it into a fraction

with our percent of £634

And that works out to be £110.95

So if we want the total cost

of the computer

We would need to add that on to our £634

plus our £110.95

Giving us a total of £744.95.

Now, if we wanted that total cost

rather than perhaps what
the VAT amount was

we could have looked at it in
a slightly different way.

We wanted 17 1/2% and we wanted the
original 100%

the computer plus the VAT

So what we could have found is

117 1/2% of £634

So we could have done

117.5 divided by 100

times by 634

Done that calculation directly
in our calculator

and come out with our answer of £744.95.

So it depends what information
you're looking for, whether you

wanted to split with the VAT or
whether you just wanted the total price.

All as we did here was add the
percentages first

before finding the percentage of
the £634

Now there's another nice way
of working out VAT

And we'll have a look at that
because you don't need a calculator

You can work it out when you're in the
middle of the shop

wherever you are. It's a good way

So let's just look at our 17 1/2% and
let's take the same example again, the £634

Well, 10%, let's work that out first.
That's a nice easy one.

We just divide by 10 so that
gives us £63.40.

5% is half of our 10%, so
that's half of £63.40.

Well, half of 60 is 30
Half of 3 is £1.50

So it's £31.30 plus the 20,
so it's £31.70.

Now 2 1/2% is half of 5%,
so if we find half of £31.70

Well, that's £15.50 + 35, so that's £15.85

10% add 5% add 2 1/2% gives us our
17 1/2%

So if we aad these up, we get 5.

8 Seven 1519
carry the 1567

gives us 310-6789
ten 11 and

there's £110.95.

Now let's look at where we're
taking a percentage off.

So far, we've been adding percentages on

Let's say we have a pair of
trainers in the shop.

And the cost of those trainers
is £75 less a 10% discount

So we want 10% of £75.
That will be the discount

So that's 10 / 100 * 75.

I'm writing it the longhand way, but
since it's 10% I can just divide by 10

and my answer would be £7.50

Of course, I then need to do the subtraction

So it will be 75 take away the £7.50

giving me £67.50 for the actual
cost of the trainers.

Let's look at that the other way

As we did with the adding
on ones we looked at it two ways

Of saying, well it's a 10% discount,
so the £75 is 100%

And we actually want 100 takeaway 10%

So we could work it out as being
90% of £75

Why we do 90 / 100 * 75 so that
we could directly come up with

our answer of £67.50 again.
It really depends whether

you're interested in knowing
what the discount is or whether

you want the final amount and
what the numbers are.

This was very easy actually to do in my
head to find 10% and take it away

And that's easy and actually working
out the 90% of my head.

So it depends on the context of your
question and the numbers involved

Let's look at a slightly
different situation now.

Where the price includes vieti
and we want to work out what the

price was before vieti.

Let's say we've got
our computer again.

But this time
it costs £699,

including Vieti.

Now, a common misconception.

Is to find 17 1/2% of
£699 and take it off.

That's incorrect because what
you're doing there is finding 17

1/2% of. A number that is
larger than the actual price

that the 17 half percent would
have been calculated on so.

Don't just find 17 1/2% of
that number if the 80s

included, we need to look
at it in a different way.

And what we say is, well,
that's 699 pounds.

Actually represents
117 1/2%.

100% of the cost of the computer
plus the 17 1/2% vieti.

So if I want to find out what it
costs without the V. 80.

What I want is the 100%.

So what we do?

Very simply, is
we divide by

117 1/2. I put it
this .5 because I'm using a

fraction. And what we have on
this side? If we divide 117 1/2

by 117 1/2 is one. So we
calculate what 1% is.

And having got our 1%.

We can then very easily
multiplied by 100 to find the

100%, so the cost before vieti.

Or if we wanted the V. 80, we
could have just multiplied by 17

1/2 to find just the V-80

Element. So the net cost
is going to be 699

/. 117.5 * 100 is
just right. The net cost.

And when we calculate that
we get £594.89 and that's

to the nearest penny.

OK, let's look at
another example.

This time we've got an insurance
company charging a customer 320

pounds. For his car insurance.

Now this includes
the government's insurance

premium tax.

A 5%.
So we want to calculate what the

cost of the car insurance is.

Before that, 5% been added on.

Well, are 320 pounds.

Represents the 100% of the cost
of the insurance plus the 5%.

Of the premium tax. So that's
100 and 5%.

So as before, we take care
320 pounds. We divide it by

105 to get 1%.

So the. Car insurance.

They thought the tax has been

added. Is going to
be our £320 /

105 * 100.

And when we calculate that.

We come up with 304 pounds.

And 76 Pence.

Let's look at another similar
example, but this time a

reduction in costs.

Let's say a coat is reduced.

In a sale.

And it's reduced by
15%.

And the new cost.

Is
£127.50.

And we want to calculate what
the original cost was.

So our £127.50? What
does that represent?

Well, it's the original 100%.

Minus the 15%.

So it's equal to
85%.

We want to calculate what the
100% was the original amount. So

if we take the £127.50.

Divided by 85.

That tells us what 1% is.

And then all this we have to do
to find the original cost.

Is to multiply our 1% by
100 so 127 pounds 50 /,

85 * 100 and if you
work that out, if we come

up with 150 pounds.

Now another percentage that
we might want to

calculate is a percentage
decrease or a percentage

increase and we've got
a rule for this.

So percentage increase or

decrease. And to calculate
it what we do is

we find the actual increase.

Or decrease.

And we divide it by
the original amount.

And then we multiply by 100%.

So we find the actual increase
or decrease divide by the

original amount and multiplied
by 100%. Let's have a look at

some examples and put it into

context. Let's say a
couple paid £180,000

for their house.

And four years later.

It was valued
at 350,000 pounds.

What's the percentage
increase?

In Valley.

Of that house.

All right, So what
percentage increase?

It's equal to the actual

increase. So

that's £350,000.
Take away 180,000 pounds. That's

the actual increase in value
divided by the original amount

will. Originally it costs them

£180,000. We multiplied by

100%. 350
takeaway 180,000

is £170,000
/ 180,000

pounds times

100%. And when you
calculate that, you find that to

the nearest percentage, that's a
94% increase in that value.

Let's have a look
at a decrease now.

Let's suppose you paid £12,000
for a car.

And three years later.

It was only worse.

8000 pounds. What
was the percentage decrease?

And the value of that car.

Inside percentage decrease.

Is equal to the actual
decrease, so that's our 12,000

pounds takeaway 8000 pounds.

Divided by the original amounts,
the original value of the car

was £12,000. And we
multiplied by 100%.

So that's £4000 /
12,000 pounds times 100%

and that gives us
33% to the nearest

1%. Let's have a look
now at using the percentage

button on your Calculator.

Now it very much depends when
you press the percentage button

in your calculation as to what
effect this has on your answer,

and I think the easiest way is
if we have a look at some

various calculations. So if I do
it in a chart with the buttons

pressed. That's what we're
actually going to press on the

Calculator. The answer

that's displayed.
The

effect. That
that's actually had. So

what it's done and what
that actually means in

terms of the calculation.

So let's look at if we press

48. Divided

by 400. And then
the percent button.

Well, the answer that's
displayed is 12.

So pressing that percent button
has had the effect of

multiplying by 100.

What does that mean in terms of
our question? Well, what we've

done is found 48.

As a percentage of
400.

Let's look at another example.

This time, if we do one.

Divided by two.

Multiplied by 300.

And then press the
percent button.

The answer that comes up is 1.5.

And the effect that this is hard
is to divide by 100.

And what we've done is, we've

found. 300%.

Of 1/2.
Let's try this

one 400.

Multiplied by 50.

And then the percent button.

The answer we
get is 200.

So this is how the effect of
dividing by 100.

And what we've done is found
50% of 400.

Now this is a common sort of
question. 50% of 400. I know

that's dead easy and you can do
it in your head, but let's

consider what happens if we
actually put that into the

Calculator. So if we do 50,
press the percent button.

And multiply by 400.

Well what's displayed another
have written answer. It's

actually what's displayed is 400
because we've not pressed any

equals, so it's actually not
calculating anything. It's had

no effect. And there's
been no calculation.

This percentage hasn't done
anything 50% * 400. You just got

that 400 in the display still.

So if we actually now press the

equals Button. Afterwards, what

happens? Well, the answer we get
then is 20,000.

Now again that pressing that
percent button, this had no

effect is what's happened is
we've just done 50 * 400.

And that's the meaning that

we've got there. So by all
means, use the percent

button on your Calculator,
but be aware of what it's

actually doing to the
calculation. Depending on

when you press it as to what
effect it has.