WEBVTT

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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

00:15:50.812 --> 00:15:52.628
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

00:16:28.410 --> 00:16:34.431
1/2% of. A number that is
larger than the actual price

00:16:34.431 --> 00:16:39.700
that the 17 half percent would
have been calculated on so.

00:16:40.330 --> 00:16:44.026
Don't just find 17 1/2% of
that number if the 80s

00:16:44.026 --> 00:16:47.722
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%.

00:17:19.590 --> 00:17:22.458
So what we do?

00:17:22.460 --> 00:17:28.496
Very simply, is
we divide by

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

00:17:34.396 --> 00:17:40.712
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.

00:17:46.900 --> 00:17:49.060
And having got our 1%.

00:17:49.670 --> 00:17:55.082
We can then very easily
multiplied by 100 to find the

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

00:18:04.594 --> 00:18:07.180
1/2 to find just the V-80

00:18:07.180 --> 00:18:14.895
Element. So the net cost
is going to be 699

00:18:14.895 --> 00:18:22.345
/. 117.5 * 100 is
just right. The net cost.

00:18:24.240 --> 00:18:31.640
And when we calculate that
we get £594.89 and that's

00:18:31.640 --> 00:18:34.600
to the nearest penny.

00:18:36.190 --> 00:18:40.528
OK, let's look at
another example.

00:18:42.030 --> 00:18:48.960
This time we've got an insurance
company charging a customer 320

00:18:48.960 --> 00:18:51.830
pounds. For his car insurance.

00:18:52.500 --> 00:18:59.568
Now this includes
the government's insurance

00:18:59.568 --> 00:19:01.924
premium tax.

00:19:02.790 --> 00:19:09.184
A 5%.
So we want to calculate what the

00:19:09.184 --> 00:19:11.476
cost of the car insurance is.

00:19:12.210 --> 00:19:14.556
Before that, 5% been added on.

00:19:16.370 --> 00:19:18.978
Well, are 320 pounds.

00:19:19.900 --> 00:19:26.836
Represents the 100% of the cost
of the insurance plus the 5%.

00:19:27.360 --> 00:19:33.336
Of the premium tax. So that's
100 and 5%.

00:19:35.050 --> 00:19:42.370
So as before, we take care
320 pounds. We divide it by

00:19:42.370 --> 00:19:44.810
105 to get 1%.

00:19:46.730 --> 00:19:50.340
So the. Car insurance.

00:19:51.830 --> 00:19:54.920
They thought the tax has been

00:19:54.920 --> 00:20:02.272
added. Is going to
be our £320 /

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105 * 100.

00:20:05.990 --> 00:20:07.680
And when we calculate that.

00:20:08.700 --> 00:20:12.510
We come up with 304 pounds.

00:20:13.040 --> 00:20:14.750
And 76 Pence.

00:20:17.750 --> 00:20:24.480
Let's look at another similar
example, but this time a

00:20:24.480 --> 00:20:26.499
reduction in costs.

00:20:27.630 --> 00:20:31.110
Let's say a coat is reduced.

00:20:33.070 --> 00:20:34.090
In a sale.

00:20:38.360 --> 00:20:42.670
And it's reduced by
15%.

00:20:44.430 --> 00:20:47.358
And the new cost.

00:20:47.360 --> 00:20:51.390
Is
£127.50.

00:20:52.510 --> 00:20:57.340
And we want to calculate what
the original cost was.

00:20:59.310 --> 00:21:05.197
So our £127.50? What
does that represent?

00:21:06.470 --> 00:21:09.160
Well, it's the original 100%.

00:21:09.670 --> 00:21:12.589
Minus the 15%.

00:21:13.630 --> 00:21:18.050
So it's equal to
85%.

00:21:21.020 --> 00:21:27.356
We want to calculate what the
100% was the original amount. So

00:21:27.356 --> 00:21:29.996
if we take the £127.50.

00:21:30.590 --> 00:21:34.178
Divided by 85.

00:21:34.180 --> 00:21:37.150
That tells us what 1% is.

00:21:38.450 --> 00:21:42.272
And then all this we have to do
to find the original cost.

00:21:43.100 --> 00:21:50.660
Is to multiply our 1% by
100 so 127 pounds 50 /,

00:21:50.660 --> 00:21:58.220
85 * 100 and if you
work that out, if we come

00:21:58.220 --> 00:22:00.740
up with 150 pounds.

00:22:01.910 --> 00:22:08.654
Now another percentage that
we might want to

00:22:08.654 --> 00:22:15.398
calculate is a percentage
decrease or a percentage

00:22:15.398 --> 00:22:22.142
increase and we've got
a rule for this.

00:22:22.142 --> 00:22:25.514
So percentage increase or

00:22:25.514 --> 00:22:32.428
decrease. And to calculate
it what we do is

00:22:32.428 --> 00:22:35.908
we find the actual increase.

00:22:37.180 --> 00:22:38.700
Or decrease.

00:22:41.670 --> 00:22:47.758
And we divide it by
the original amount.

00:22:50.180 --> 00:22:53.636
And then we multiply by 100%.

00:22:54.750 --> 00:22:59.491
So we find the actual increase
or decrease divide by the

00:22:59.491 --> 00:23:04.232
original amount and multiplied
by 100%. Let's have a look at

00:23:04.232 --> 00:23:06.818
some examples and put it into

00:23:06.818 --> 00:23:13.286
context. Let's say a
couple paid £180,000

00:23:13.286 --> 00:23:16.019
for their house.

00:23:17.300 --> 00:23:20.200
And four years later.

00:23:20.200 --> 00:23:27.886
It was valued
at 350,000 pounds.

00:23:27.930 --> 00:23:32.238
What's the percentage
increase?

00:23:33.360 --> 00:23:34.550
In Valley.

00:23:35.640 --> 00:23:37.518
Of that house.

00:23:38.470 --> 00:23:42.376
All right, So what
percentage increase?

00:23:43.570 --> 00:23:46.375
It's equal to the actual

00:23:46.375 --> 00:23:49.246
increase. So

00:23:49.246 --> 00:23:56.000
that's £350,000.
Take away 180,000 pounds. That's

00:23:56.000 --> 00:24:02.460
the actual increase in value
divided by the original amount

00:24:02.460 --> 00:24:05.690
will. Originally it costs them

00:24:05.690 --> 00:24:09.605
£180,000. We multiplied by

00:24:09.605 --> 00:24:15.840
100%. 350
takeaway 180,000

00:24:15.840 --> 00:24:22.160
is £170,000
/ 180,000

00:24:22.160 --> 00:24:25.320
pounds times

00:24:25.320 --> 00:24:32.481
100%. And when you
calculate that, you find that to

00:24:32.481 --> 00:24:37.871
the nearest percentage, that's a
94% increase in that value.

00:24:38.410 --> 00:24:45.498
Let's have a look
at a decrease now.

00:24:45.630 --> 00:24:51.638
Let's suppose you paid £12,000
for a car.

00:24:52.640 --> 00:24:56.420
And three years later.

00:24:57.410 --> 00:24:59.198
It was only worse.

00:25:00.080 --> 00:25:06.010
8000 pounds. What
was the percentage decrease?

00:25:07.220 --> 00:25:09.296
And the value of that car.

00:25:11.140 --> 00:25:14.119
Inside percentage decrease.

00:25:15.770 --> 00:25:23.240
Is equal to the actual
decrease, so that's our 12,000

00:25:23.240 --> 00:25:26.228
pounds takeaway 8000 pounds.

00:25:26.280 --> 00:25:32.693
Divided by the original amounts,
the original value of the car

00:25:32.693 --> 00:25:38.460
was £12,000. And we
multiplied by 100%.

00:25:38.790 --> 00:25:46.678
So that's £4000 /
12,000 pounds times 100%

00:25:46.678 --> 00:25:54.566
and that gives us
33% to the nearest

00:25:54.566 --> 00:26:00.844
1%. Let's have a look
now at using the percentage

00:26:00.844 --> 00:26:02.908
button on your Calculator.

00:26:05.490 --> 00:26:10.693
Now it very much depends when
you press the percentage button

00:26:10.693 --> 00:26:16.369
in your calculation as to what
effect this has on your answer,

00:26:16.369 --> 00:26:22.991
and I think the easiest way is
if we have a look at some

00:26:22.991 --> 00:26:29.501
various calculations. So if I do
it in a chart with the buttons

00:26:29.501 --> 00:26:32.749
pressed. That's what we're
actually going to press on the

00:26:32.749 --> 00:26:35.780
Calculator. The answer

00:26:35.780 --> 00:26:40.980
that's displayed.
The

00:26:40.980 --> 00:26:46.670
effect. That
that's actually had. So

00:26:46.670 --> 00:26:51.350
what it's done and what
that actually means in

00:26:51.350 --> 00:26:53.430
terms of the calculation.

00:26:54.660 --> 00:26:58.594
So let's look at if we press

00:26:58.594 --> 00:27:01.466
48. Divided

00:27:01.466 --> 00:27:06.280
by 400. And then
the percent button.

00:27:08.200 --> 00:27:11.728
Well, the answer that's
displayed is 12.

00:27:12.350 --> 00:27:18.430
So pressing that percent button
has had the effect of

00:27:18.430 --> 00:27:20.254
multiplying by 100.

00:27:21.100 --> 00:27:27.088
What does that mean in terms of
our question? Well, what we've

00:27:27.088 --> 00:27:29.084
done is found 48.

00:27:29.680 --> 00:27:34.380
As a percentage of
400.

00:27:36.170 --> 00:27:38.290
Let's look at another example.

00:27:38.830 --> 00:27:41.446
This time, if we do one.

00:27:42.530 --> 00:27:45.239
Divided by two.

00:27:45.240 --> 00:27:49.239
Multiplied by 300.

00:27:49.240 --> 00:27:50.818
And then press the
percent button.

00:27:52.390 --> 00:27:55.729
The answer that comes up is 1.5.

00:27:57.040 --> 00:28:03.448
And the effect that this is hard
is to divide by 100.

00:28:05.030 --> 00:28:07.274
And what we've done is, we've

00:28:07.274 --> 00:28:10.710
found. 300%.

00:28:11.650 --> 00:28:19.358
Of 1/2.
Let's try this

00:28:19.358 --> 00:28:21.670
one 400.

00:28:23.250 --> 00:28:27.138
Multiplied by 50.

00:28:27.150 --> 00:28:28.930
And then the percent button.

00:28:29.540 --> 00:28:34.430
The answer we
get is 200.

00:28:35.930 --> 00:28:41.000
So this is how the effect of
dividing by 100.

00:28:41.570 --> 00:28:47.213
And what we've done is found
50% of 400.

00:28:49.740 --> 00:28:54.316
Now this is a common sort of
question. 50% of 400. I know

00:28:54.316 --> 00:28:58.892
that's dead easy and you can do
it in your head, but let's

00:28:58.892 --> 00:29:02.412
consider what happens if we
actually put that into the

00:29:02.412 --> 00:29:05.932
Calculator. So if we do 50,
press the percent button.

00:29:06.490 --> 00:29:09.378
And multiply by 400.

00:29:11.790 --> 00:29:15.814
Well what's displayed another
have written answer. It's

00:29:15.814 --> 00:29:20.844
actually what's displayed is 400
because we've not pressed any

00:29:20.844 --> 00:29:25.371
equals, so it's actually not
calculating anything. It's had

00:29:25.371 --> 00:29:30.450
no effect. And there's
been no calculation.

00:29:31.790 --> 00:29:36.564
This percentage hasn't done
anything 50% * 400. You just got

00:29:36.564 --> 00:29:39.168
that 400 in the display still.

00:29:40.410 --> 00:29:44.106
So if we actually now press the

00:29:44.106 --> 00:29:47.576
equals Button. Afterwards, what

00:29:47.576 --> 00:29:52.710
happens? Well, the answer we get
then is 20,000.

00:29:54.160 --> 00:29:59.180
Now again that pressing that
percent button, this had no

00:29:59.180 --> 00:30:04.702
effect is what's happened is
we've just done 50 * 400.

00:30:04.820 --> 00:30:06.355
And that's the meaning that

00:30:06.355 --> 00:30:10.886
we've got there. So by all
means, use the percent

00:30:10.886 --> 00:30:15.166
button on your Calculator,
but be aware of what it's

00:30:15.166 --> 00:30:18.162
actually doing to the
calculation. Depending on

00:30:18.162 --> 00:30:22.442
when you press it as to what
effect it has.