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.