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

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

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And before we finish will look at

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how to use the percentage
button on your calculator.

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Now, percentages are very
familiar to us.

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We see them in the media an 
awful lot,

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whether they're talking about the

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percentage of A to Cs
gained by youngsters in exams,

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

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medical procedures.

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

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

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The National Insurance
contribution is

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a percentage of your salary.

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

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

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

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So what does percentage
actually mean?

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

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And it means
out of 100.

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

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So let's have a look at an
example.

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

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

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Let's look at some
familiar percentages

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75%
that means 75 out of 100.

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

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Or if we write it
as a decimal 0.75

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50%
That's 50 out of 100.

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

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and as a decimal 0.5

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25% is 25 out of 100

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In its lowest form, that's a quarter.

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or 0.25 as a decimal

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

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That's 10 out of 100

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or 1/10 as a fraction in its lowest form

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or 0.1 as a decimal

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

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

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and 0.05 as a decimal.

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Now it's worth noting here that

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to find 50% an easy way of doing it

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is just to find half of the number

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or divide by two.

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And to find 10% another common one

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You just divide by 10,

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finding a tenth of the number.

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

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

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and we want to write that as a percentage.

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First of all, we write it as a fraction:

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18 out of 20

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

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the denominator to be 100.

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Well 20 multiplied by 5 gives us 100

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So in this case

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we can just multiply 18 by 5.

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Which gives us 90 out of 100.

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So 18 out of 20 is the same as 90%

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

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

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

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Well, let's have a look.

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Let's take the example this time

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Let's say we have 53 marks.

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

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

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And what we're going to do

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here we multiplied by 5 and

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we multiplied the denominator by 5.

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So both numerator and denominator were 
multiplied by 5

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

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Now this 100 here gives us
our percentage because

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percentage is divide by 100.

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

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multiply it by 100.

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And that will give us our percentage.

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And if we carry out that calculation,

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we find that it comes to 77.94%

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That's to two decimal places.

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Or if we put it to the nearest percent
78%

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

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finding the percentage of a quantity,

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

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So let's say our meal costs £25.40

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and we want to leave a 10% tip

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Now we want 10% of £25.40

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So in mathematical language

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10% is 10 divided by 100

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'of' is multiply

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£25.40

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And that equals £2.54,

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so that's the tip that we would leave

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Now 10% I said earlier that was an easy

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one to calculate because you just

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divide by 10.

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But the method that I've shown here

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is what you would use for

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more complex percentages.

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Let's have a look at another one.

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

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on orders that he gets in one month.

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

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How much is he going to earn?

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Well, we want 2% of £250,000

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Again turn to our mathematical language

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2% is 2 divided by 100

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'of' (multiply) £250,000

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Now in this one we can do some cancelling

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because 100 goes into £250,000

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So we've got twice 2500

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Giving us an answer of £5000.

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

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So let's do some calculations where we

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need to add on VAT

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Let's say a computer costs

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£634 plus VAT

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

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We can find 17 and a half percent of £634

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

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Well, let's do that.

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So 17 1/2% of £634 is 17.5

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I'm going to write it as .5

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when I now turn it into a fraction

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with our percent of £634

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And that works out to be £110.95

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So if we want the total cost

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of the computer

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We would need to add that on to our £634

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plus our £110.95

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Giving us a total of £744.95.

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Now, if we wanted that total cost

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rather than perhaps what 
the VAT amount was

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

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

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the computer plus the VAT

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So what we could have found is

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117 1/2% of £634

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So we could have done

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117.5 divided by 100

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times by 634

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Done that calculation directly 
in our calculator

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and come out with our answer of £744.95.

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

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

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All as we did here was add the 
percentages first

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before finding the percentage of
the £634

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Now there's another nice way
of working out VAT

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

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

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wherever you are. It's a good way

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

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

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

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

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

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

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

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

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

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So if we aad these up, we get 5.

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8 Seven 1519
carry the 1567

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gives us 310-6789
ten 11 and

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there's £110.95.

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

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So far, we've been adding percentages on

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

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

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

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So that's 10 / 100 * 75.

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

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and my answer would be £7.50

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Of course, I then need to do the subtraction

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So it will be 75 take away the £7.50

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

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Let's look at that the other way

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

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

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And we actually want 100 takeaway 10%

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

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

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our answer of £67.50 again.
It really depends whether

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

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you want the final amount and
what the numbers are.

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

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

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

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Let's look at a slightly
different situation now.

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

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price was before vieti.

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

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But this time
it costs £699,

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including Vieti.

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Now, a common misconception.

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

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

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

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

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

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

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

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Actually represents
117 1/2%.

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

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

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What I want is the 100%.

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So what we do?

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Very simply, is
we divide by

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

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

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

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And having got our 1%.

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00:17:49,670 --> 00:17:55,082
We can then very easily
multiplied by 100 to find the

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00:17:55,082 --> 00:17:58,034
100%, so the cost before vieti.

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

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1/2 to find just the V-80

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Element. So the net cost
is going to be 699

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00:18:14,895 --> 00:18:22,345
/. 117.5 * 100 is
just right. The net cost.

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

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to the nearest penny.

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OK, let's look at
another example.

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

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pounds. For his car insurance.

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Now this includes
the government's insurance

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premium tax.

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A 5%.
So we want to calculate what the

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cost of the car insurance is.

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Before that, 5% been added on.

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Well, are 320 pounds.

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00:19:19,900 --> 00:19:26,836
Represents the 100% of the cost
of the insurance plus the 5%.

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00:19:27,360 --> 00:19:33,336
Of the premium tax. So that's
100 and 5%.

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00:19:35,050 --> 00:19:42,370
So as before, we take care
320 pounds. We divide it by

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105 to get 1%.

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So the. Car insurance.

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They thought the tax has been

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added. Is going to
be our £320 /

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00:20:02,272 --> 00:20:04,960
105 * 100.

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00:20:05,990 --> 00:20:07,680
And when we calculate that.

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We come up with 304 pounds.

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And 76 Pence.

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00:20:17,750 --> 00:20:24,480
Let's look at another similar
example, but this time a

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reduction in costs.

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00:20:27,630 --> 00:20:31,110
Let's say a coat is reduced.

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In a sale.

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00:20:38,360 --> 00:20:42,670
And it's reduced by
15%.

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00:20:44,430 --> 00:20:47,358
And the new cost.

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00:20:47,360 --> 00:20:51,390
Is
£127.50.

248
00:20:52,510 --> 00:20:57,340
And we want to calculate what
the original cost was.

249
00:20:59,310 --> 00:21:05,197
So our £127.50? What
does that represent?

250
00:21:06,470 --> 00:21:09,160
Well, it's the original 100%.

251
00:21:09,670 --> 00:21:12,589
Minus the 15%.

252
00:21:13,630 --> 00:21:18,050
So it's equal to
85%.

253
00:21:21,020 --> 00:21:27,356
We want to calculate what the
100% was the original amount. So

254
00:21:27,356 --> 00:21:29,996
if we take the £127.50.

255
00:21:30,590 --> 00:21:34,178
Divided by 85.

256
00:21:34,180 --> 00:21:37,150
That tells us what 1% is.

257
00:21:38,450 --> 00:21:42,272
And then all this we have to do
to find the original cost.

258
00:21:43,100 --> 00:21:50,660
Is to multiply our 1% by
100 so 127 pounds 50 /,

259
00:21:50,660 --> 00:21:58,220
85 * 100 and if you
work that out, if we come

260
00:21:58,220 --> 00:22:00,740
up with 150 pounds.

261
00:22:01,910 --> 00:22:08,654
Now another percentage that
we might want to

262
00:22:08,654 --> 00:22:15,398
calculate is a percentage
decrease or a percentage

263
00:22:15,398 --> 00:22:22,142
increase and we've got
a rule for this.

264
00:22:22,142 --> 00:22:25,514
So percentage increase or

265
00:22:25,514 --> 00:22:32,428
decrease. And to calculate
it what we do is

266
00:22:32,428 --> 00:22:35,908
we find the actual increase.

267
00:22:37,180 --> 00:22:38,700
Or decrease.

268
00:22:41,670 --> 00:22:47,758
And we divide it by
the original amount.

269
00:22:50,180 --> 00:22:53,636
And then we multiply by 100%.

270
00:22:54,750 --> 00:22:59,491
So we find the actual increase
or decrease divide by the

271
00:22:59,491 --> 00:23:04,232
original amount and multiplied
by 100%. Let's have a look at

272
00:23:04,232 --> 00:23:06,818
some examples and put it into

273
00:23:06,818 --> 00:23:13,286
context. Let's say a
couple paid £180,000

274
00:23:13,286 --> 00:23:16,019
for their house.

275
00:23:17,300 --> 00:23:20,200
And four years later.

276
00:23:20,200 --> 00:23:27,886
It was valued
at 350,000 pounds.

277
00:23:27,930 --> 00:23:32,238
What's the percentage
increase?

278
00:23:33,360 --> 00:23:34,550
In Valley.

279
00:23:35,640 --> 00:23:37,518
Of that house.

280
00:23:38,470 --> 00:23:42,376
All right, So what
percentage increase?

281
00:23:43,570 --> 00:23:46,375
It's equal to the actual

282
00:23:46,375 --> 00:23:49,246
increase. So

283
00:23:49,246 --> 00:23:56,000
that's £350,000.
Take away 180,000 pounds. That's

284
00:23:56,000 --> 00:24:02,460
the actual increase in value
divided by the original amount

285
00:24:02,460 --> 00:24:05,690
will. Originally it costs them

286
00:24:05,690 --> 00:24:09,605
£180,000. We multiplied by

287
00:24:09,605 --> 00:24:15,840
100%. 350
takeaway 180,000

288
00:24:15,840 --> 00:24:22,160
is £170,000
/ 180,000

289
00:24:22,160 --> 00:24:25,320
pounds times

290
00:24:25,320 --> 00:24:32,481
100%. And when you
calculate that, you find that to

291
00:24:32,481 --> 00:24:37,871
the nearest percentage, that's a
94% increase in that value.

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00:24:38,410 --> 00:24:45,498
Let's have a look
at a decrease now.

293
00:24:45,630 --> 00:24:51,638
Let's suppose you paid £12,000
for a car.

294
00:24:52,640 --> 00:24:56,420
And three years later.

295
00:24:57,410 --> 00:24:59,198
It was only worse.

296
00:25:00,080 --> 00:25:06,010
8000 pounds. What
was the percentage decrease?

297
00:25:07,220 --> 00:25:09,296
And the value of that car.

298
00:25:11,140 --> 00:25:14,119
Inside percentage decrease.

299
00:25:15,770 --> 00:25:23,240
Is equal to the actual
decrease, so that's our 12,000

300
00:25:23,240 --> 00:25:26,228
pounds takeaway 8000 pounds.

301
00:25:26,280 --> 00:25:32,693
Divided by the original amounts,
the original value of the car

302
00:25:32,693 --> 00:25:38,460
was £12,000. And we
multiplied by 100%.

303
00:25:38,790 --> 00:25:46,678
So that's £4000 /
12,000 pounds times 100%

304
00:25:46,678 --> 00:25:54,566
and that gives us
33% to the nearest

305
00:25:54,566 --> 00:26:00,844
1%. Let's have a look
now at using the percentage

306
00:26:00,844 --> 00:26:02,908
button on your Calculator.

307
00:26:05,490 --> 00:26:10,693
Now it very much depends when
you press the percentage button

308
00:26:10,693 --> 00:26:16,369
in your calculation as to what
effect this has on your answer,

309
00:26:16,369 --> 00:26:22,991
and I think the easiest way is
if we have a look at some

310
00:26:22,991 --> 00:26:29,501
various calculations. So if I do
it in a chart with the buttons

311
00:26:29,501 --> 00:26:32,749
pressed. That's what we're
actually going to press on the

312
00:26:32,749 --> 00:26:35,780
Calculator. The answer

313
00:26:35,780 --> 00:26:40,980
that's displayed.
The

314
00:26:40,980 --> 00:26:46,670
effect. That
that's actually had. So

315
00:26:46,670 --> 00:26:51,350
what it's done and what
that actually means in

316
00:26:51,350 --> 00:26:53,430
terms of the calculation.

317
00:26:54,660 --> 00:26:58,594
So let's look at if we press

318
00:26:58,594 --> 00:27:01,466
48. Divided

319
00:27:01,466 --> 00:27:06,280
by 400. And then
the percent button.

320
00:27:08,200 --> 00:27:11,728
Well, the answer that's
displayed is 12.

321
00:27:12,350 --> 00:27:18,430
So pressing that percent button
has had the effect of

322
00:27:18,430 --> 00:27:20,254
multiplying by 100.

323
00:27:21,100 --> 00:27:27,088
What does that mean in terms of
our question? Well, what we've

324
00:27:27,088 --> 00:27:29,084
done is found 48.

325
00:27:29,680 --> 00:27:34,380
As a percentage of
400.

326
00:27:36,170 --> 00:27:38,290
Let's look at another example.

327
00:27:38,830 --> 00:27:41,446
This time, if we do one.

328
00:27:42,530 --> 00:27:45,239
Divided by two.

329
00:27:45,240 --> 00:27:49,239
Multiplied by 300.

330
00:27:49,240 --> 00:27:50,818
And then press the
percent button.

331
00:27:52,390 --> 00:27:55,729
The answer that comes up is 1.5.

332
00:27:57,040 --> 00:28:03,448
And the effect that this is hard
is to divide by 100.

333
00:28:05,030 --> 00:28:07,274
And what we've done is, we've

334
00:28:07,274 --> 00:28:10,710
found. 300%.

335
00:28:11,650 --> 00:28:19,358
Of 1/2.
Let's try this

336
00:28:19,358 --> 00:28:21,670
one 400.

337
00:28:23,250 --> 00:28:27,138
Multiplied by 50.

338
00:28:27,150 --> 00:28:28,930
And then the percent button.

339
00:28:29,540 --> 00:28:34,430
The answer we
get is 200.

340
00:28:35,930 --> 00:28:41,000
So this is how the effect of
dividing by 100.

341
00:28:41,570 --> 00:28:47,213
And what we've done is found
50% of 400.

342
00:28:49,740 --> 00:28:54,316
Now this is a common sort of
question. 50% of 400. I know

343
00:28:54,316 --> 00:28:58,892
that's dead easy and you can do
it in your head, but let's

344
00:28:58,892 --> 00:29:02,412
consider what happens if we
actually put that into the

345
00:29:02,412 --> 00:29:05,932
Calculator. So if we do 50,
press the percent button.

346
00:29:06,490 --> 00:29:09,378
And multiply by 400.

347
00:29:11,790 --> 00:29:15,814
Well what's displayed another
have written answer. It's

348
00:29:15,814 --> 00:29:20,844
actually what's displayed is 400
because we've not pressed any

349
00:29:20,844 --> 00:29:25,371
equals, so it's actually not
calculating anything. It's had

350
00:29:25,371 --> 00:29:30,450
no effect. And there's
been no calculation.

351
00:29:31,790 --> 00:29:36,564
This percentage hasn't done
anything 50% * 400. You just got

352
00:29:36,564 --> 00:29:39,168
that 400 in the display still.

353
00:29:40,410 --> 00:29:44,106
So if we actually now press the

354
00:29:44,106 --> 00:29:47,576
equals Button. Afterwards, what

355
00:29:47,576 --> 00:29:52,710
happens? Well, the answer we get
then is 20,000.

356
00:29:54,160 --> 00:29:59,180
Now again that pressing that
percent button, this had no

357
00:29:59,180 --> 00:30:04,702
effect is what's happened is
we've just done 50 * 400.

358
00:30:04,820 --> 00:30:06,355
And that's the meaning that

359
00:30:06,355 --> 00:30:10,886
we've got there. So by all
means, use the percent

360
00:30:10,886 --> 00:30:15,166
button on your Calculator,
but be aware of what it's

361
00:30:15,166 --> 00:30:18,162
actually doing to the
calculation. Depending on

362
00:30:18,162 --> 00:30:22,442
when you press it as to what
effect it has.