## www.mathcentre.ac.uk/.../Ratios.mp4

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Today's session is on ratio.
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I'm going to explain what a ratio is
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and how ratios are used
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in different situations.
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So to start off with what is a ratio?
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Well, a ratio is a way of comparing
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amounts of ingredients.
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Ratios can be used to compare
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weights, money, length and so on.
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So if we take this example
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we've got a model boat
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whose length is 1 metre
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and the real boat
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whose length is 25 metres.
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Then we say the ratio of the
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length of the model boat to the real boat
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is 1 to 25.
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Notice we've just used the numbers
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without the unit (metres)
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and we've used the colon
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to represent the ratio.
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Ratios are used to describe quantities
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of ingredients in mixtures.
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For example, in the
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when you're making medicines,
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when you are making cement or mortar,
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or at home when you're making up food
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you use different quantities
in different proportions
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and if you don't get them right
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then things go wrong.
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So it's very important to know
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what quantities you've got and
in what ratio.
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So for example, if we have
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mortar for building brick walls.
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Mortar is made up by mixing
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two parts of cement to
seven parts of gravel by volume
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and we write that ratio as 2 to 7.
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Again notice we've used
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the numbers without the units
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and the colon to represent the ratio.
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When we're making pastry at home,
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when we're making pies and tarts,
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we mix four ounces of flour
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with two ounces of margarine
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And that ratio would be 4 to 2.
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But in this case,
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if you look at the numbers,
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they've got a factor of two in common
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So we can simplify ratios just in the same
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way as we simplify fractions.
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We can divide by the common factor,
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so we divide 4 by 2
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and 2 by 2 to give 1.
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So 2 to 1 is the simplest form
of the ratio 4 to 2.
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But both of the ratios are equivalent,
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because the relationship of the numbers
involved stays the same.
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If we take this example
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250 to 150
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We can simplify this ratio.
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We divide both by 10
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to get 25 to 15
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And then we can divide both by 5
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5 into 25 will give me 5
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5 into 15 will give me 3
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We can't divide anymore,
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so this is the simplest form.
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5 to 3 the simplest form of
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the ratio 250 to 150.
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But all three ratios are equivalent
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because the relationship of the numbers
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is exactly the same.
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In the same way, we can actually
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simplify this ratio: 1 to 1.5
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In ratios we like to have whole numbers
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and in this ratio you can see
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that we have a decimal.
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To get rid of the decimal
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we can multiply both sides of the
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ratio by 10
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and we still have an equivalent ratio.
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Because, again, the relationship
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between the numbers is the same.
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So we multiply the 1 by 10
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you get 10
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Multiply 1.5 by 10
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you get 15
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10 to 15 we can simplify that.
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Divide both sides by 5.
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5 into 10 gives me 2
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5 into 15 will give me 3
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2 to 3 is the simplest form
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of the ratio 1 to 1.5
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Similarly, when we have fractions
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a quarter to five-eighths,
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it just doesn't look right.
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We want to express that ratio
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in terms of whole numbers
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in its simplest form.
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So what we do first is we write
both as fractions over 8
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in terms of eighths.
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A quarter is two eighths
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and now the ratio is two eighths to five eighths.
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All we have to say is that is 2 to 5.
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We multiply both ratios by 8
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And 2 to 5 is the simplest ratio
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for the ratio a quarter to five-eighths.
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But again all three ratios are
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equivalent because the relationship
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between the numbers is exactly the same.
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Moving on,
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we must have the numbers
in the ratios
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having the same units.
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So if we have this ratio
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15 pence to 3 pounds,
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we cannot say that the ratio is 15 to 3
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and then simplify that to 5 to 1
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Because we didn't start off with
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the numbers having the same units
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the relationship between
the numbers is not the same,
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because as I say,
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we didn't start off with these numbers
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having the same units.
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So we must convert the numbers
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to the same units
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and we choose whichever unit is
appropriate
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In this case, it's obvious we must
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change them to pence.
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So we say the ratio is 15 to 300
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as there's 300 pence for 3 pounds
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and then we simplify as normal.
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We divide both sides by five.
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5 into 15 is 3
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5 into 300 is 60
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And then we can divide by 3
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3 into 3 is 1
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3 into 60 is 20
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And notice these two ratios are
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not the same, they're vastly different.
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They're not equivalent because
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the relationship between the numbers
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is not the same.
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So it's very important in ratios
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that have the same units.
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If they're not,
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then you convert them to the same units
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and then simplify if appropriate.
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As I said before,
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ratios are extremely useful
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in lots of different circumstances.
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They can be used to divide and
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share amounts of different quantities
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like money, weights, and so on.
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So if I take this problem
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just say I had an inheritance of £64,000
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and it was to be shared between two people
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Mrs Sharp and Mr West
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in the ratio 5 to 3
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What I want you to do is work out
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what each one of those gets.
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And that's a lot of information to take in
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so what I do first is
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I start off with a diagram
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I've got the total inheritance of £64,000
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and I divide it
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between Mrs Sharp
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and Mr West
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in the ratio 5 to 3
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And we want to work out
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what each gets.
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What we do first is we work out
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the total number of parts that
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their inheritance is split up into.
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Well, we use the ratio for that.
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It's five parts for Mrs Sharp
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and three parts for Mr West
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so altogether that is eight parts
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Then we work out what the total value
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of one part of the inheritance would be.
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Now we know that the total inheritance
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is £64,000
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so one part
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equals 64,000 divided by 8
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and that is £8000
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And then the rest is easy.
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All we have to do now is
take Mrs Sharp
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and she has five parts
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and that is 5 multiplied by £8000
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which works out to be £40,000
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And then Mr West
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he has three parts
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and that is 3 multiplied by £8000
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which is £24,000
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An awful lot of money!
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But what if I made a mistake?
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How can I check my two answers?
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How can I check that Mrs Sharp did get
£40,000
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and Mr West got £24,000?
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Well a very simple check
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is to add up these two values
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to make up the total inheritance
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then we think we've done our
calculations properly.
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So a quick check:
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£40,000 plus 24,000 does equal £64,000
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For a complete check though
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we can take the two amounts
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and see that they will actually make an
equivalent ratio
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to the ratio that we started off with 5:3
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So if we take our 40,000 that
Mrs Sharp got
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and then the 24,000 that Mr West got
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and cancel them down,
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we cancel by 1000
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then we cancel by 4
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so that would make 10 to 6
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and then cancel by 2
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so that will make 5 to 3
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We do actually get the same ratio
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that we started off with.
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We're going to do another example.
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It's an example which involves another
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mixture: making concrete.
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And with this, concrete is made by mixing
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gravel, sand and cement
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in the ratio 3 to 2 to 1
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and in this problem we
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The amount of concrete
that we are going to make
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will be 12 cubic metres.
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And what I want to work out
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is how much gravel will be needed
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to make 12 cubic metres of concrete.
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and that represents the concrete
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and we know we want to make
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12 cubic metres of concrete
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and we know it's mixed
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by mixing gravel, sand and cement
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in the ratio 3 to 2 to 1
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And we want to work out
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the amount of concrete for 12 cubic metres
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Well, first of all,
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we work out the total number of parts
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our concrete is divided up into
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and we use our ratio for that.
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It's 3 + 2 + 1 and that equals 6 parts
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Now our concrete is divided up
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into six parts
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So one part must equal
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our 12 cubic metres divided by 6
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so that's 12 divided by 6 cubic metres
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which works out to be 2 cubic metres.
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Now we want to work out
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how much gravel is needed.
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Gravel is represented by 3 parts
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so gravel, the amount that we want
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equals 3 times 2 cubic metres
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which is 6 cubic metres
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But it's always good to check
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and so we try and do the calculation
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in a different way
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and the way that I'd like to do it
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is using fractions.
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If we go back to the original diagram
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we know that gravel is represented
by 3 parts
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and the total is 6
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so gravel is a half of
the total volume
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and a half of 12 cubic metres is
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6 cubic metres
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we've done a check.
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But what if we did a similar problem
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and we want to start off
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with mixing our concrete
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using gravel, sand, and cement
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but we don't know the final volume of
the concrete
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but we do know that we are given
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6 cubic metres of sand
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and an unlimited supply of
gravel and cement.
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How much concrete can we make then
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if we've got 6 cubic metres of sand?
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Alright, we'll start the
question or the problem
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with a diagram.
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We know that the mixture is
still the same.
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We use the same ratio
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gravel to sand to cement
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as 3 to 2 to 1
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And we know that
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we have 6 cubic metres of sand
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but we want to work out
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how much concrete we can make
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with that amount of sand
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and unlimited amounts of the other two.
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Well, the number of parts that
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the concrete is divided up into is still 6
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But we know that 2 parts
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is 6 cubic metres
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because that's what we're given
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so 2 parts equals 6 cubic metres.
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So 1 part
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equals 6 divided by 2
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which is 3 cubic metres
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Now the total number of parts of
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the concrete is divided up into is 6
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So the amount of concrete that is produced
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is 6 times 3 cubic metres
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and that is 18 cubic metres
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Again, it's good to check our answer
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and we'll do it in a different way
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and we'll use fractions again this time.
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We look at what we were given.
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Sand is represented by 2 parts
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and we know it has a volume
of 6 cubic metres.
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Altogether, there are 6 parts
for our concrete.
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So the fraction that represents sand
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is 2 over 6, which is a third.
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So a third of the total amount is
6 cubic metres
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So the whole amount of concrete must be
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3 times 6 cubic metres
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which is 18 cubic metres
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Here's another ratio problem involved
with ingredients
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but this time the ingredients are to make
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the Greek food houmous.
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It's usually given as a starter
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and there are four ingredients:
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two cloves of garlic
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are combined with
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four ounces of chickpeas
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and four tablespoonfuls
of olive oil.
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I sound a little bit like
Delia Smith at this point
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and the final secret ingredient is
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the 5 fluid ounces of tahini paste.
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Now when you combine these ingredients
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together that's enough for six people
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But what if I want to make houmous
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for nine people?
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What amounts do I have of these four
ingredients
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to make it for nine people?
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Well, we start off with what we've got
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and what we know
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We've got 2 cloves of garlic
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with 4 ounces of chickpeas
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4 tablespoonsful of olive oil
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and 5 fluid ounces of tahini paste
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and that makes enough for six people
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What I do next is that I work out
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what each of those ingredients
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would be for one person.
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So I have to divide
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each of those numbers by 6
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So that's 2 over 6
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4 over 6
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4 over 6
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and 5 over 6
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and then we cancel down if we can
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In this case we can
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that's one third.
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Cancel four sixths to two thirds.
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And this will be the same.
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And the last one just remains the same:
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five sixths
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And now it's dead easy to work out
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what amounts we need for nine people.
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All we have to do is multiply by 9
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So that's 1/3 multiplied by 9
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2/3 multiplied by 9
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and another 2/3 multiplied by 9
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and then 5/6 multiplied by 9
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And we work out these
calculations and simplify
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3 into 9 is 3
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3 into 9 is 3
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and then 2 threes are 6.
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and this works out to be the same
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which is 6 because it's the same
calculation
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3 into 6 is 2
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3 into 9 is 3
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5 threes are 15 over 2
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which works out to be 7 and a half
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for the ingredients
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is 3 cloves of garlic
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6 ounces of chickpeas
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combined with 6 tablespoonfuls
of olive oil
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and 7 and a half fluid ounces
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of tahini paste
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And that makes enough
houmous for nine people.
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In a similar way,
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you can use this method in conversion
problems
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If we had the conversion that
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1 pound is the same as 1.65 euros
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and I wanted to work out
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what 50 euros would be in pence
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to the nearest pence
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What I like doing first is to work out
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what 1 euro is in terms of pence
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1.65 euros equals 100 pence
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One euro would then equal
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100 divided by the 1.65
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And then to work out
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what the 50 euros would be
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I multiply this by 50
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as 100 over 1.65 multiplied by 50
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And that is 5000
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divided by the 1.65
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Now I am not going to do this by
long division.
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I'll use my calculator
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and I just type in the relevant numbers
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5000 divided by 1.65
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equals
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3030 point 3 0 point 3 0 repeating
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So 50 euros equals 3030 pence
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to the nearest pence.
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Which is 30 pounds and 30p
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Well, that's the session finished
now on ratio.
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Before I finish finally,
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what I'd like to do is just remind you
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of a few key points about ratio.
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First of all, what is a ratio?
• 23:52 - 23:54
Well a ratio is a way of comparing
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quantities of a similar type
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When you write a ratio down
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you use whole numbers
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separated by colon.
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The numbers should be in the
same units.
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If they're not, you convert them
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to the same units
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by using one or the other of the
units involved
• 24:14 - 24:17
• 24:17 - 24:21
And then you simplify as appropriate.
• 24:21 - 24:23
In calculations involved in ratio
• 24:23 - 24:30
it is useful to work out the total
number of parts
• 24:30 - 24:33
the quantity is divided up into
• 24:33 - 24:37
and then work out one part represents.
Title:
www.mathcentre.ac.uk/.../Ratios.mp4
Video Language:
English
Duration:
24:43
 jn17486 edited English subtitles for www.mathcentre.ac.uk/.../Ratios.mp4 jn17486 edited English subtitles for www.mathcentre.ac.uk/.../Ratios.mp4 jn17486 edited English subtitles for www.mathcentre.ac.uk/.../Ratios.mp4 jn17486 edited English subtitles for www.mathcentre.ac.uk/.../Ratios.mp4 jn17486 edited English subtitles for www.mathcentre.ac.uk/.../Ratios.mp4 jn17486 edited English subtitles for www.mathcentre.ac.uk/.../Ratios.mp4 jn17486 edited English subtitles for www.mathcentre.ac.uk/.../Ratios.mp4 jn17486 edited English subtitles for www.mathcentre.ac.uk/.../Ratios.mp4

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