The basics of ratio

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The basics of ratio
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• The purpose of this staff tutorial is to
  introduce the basic ideas of ratio and to show
  how ratio relates to fractions. Using this as a
  basis you should probably then move on to the
  tutorials Breakfast in Sydney and Orange and
  Water, where we look applications of ratio to
  exchange rates and proportions, respectively.

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• Have a look at the rectangle below.
• Its pretty easy to see that one-third of the
  rectangle is red and two-thirds is yellow. So
  there is twice as much yellow as red. So we say
  that yellow to red are in the ratio of 21.
• Ratios are always represented by a colon (as in
  21) or by the word to (as in 2 to 1). In fact
  we read 21 as two to one.
• Two to one simply means that there is twice as
  much of one thing as there is of the other.

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• Lets have a look at another example. In the
  rectangle below, there is three times as much
  green as there is blue. So what is the ratio of
  green to blue?
• How much more of green is there than of blue?

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• Because there is three times as much green as
  blue the ratio of green to blue is 31.
• So when we have simple multiples like this there
  is no problem with finding the ratios. But here
  is one with a little variation. The rectangle
  below is 3/5 green and 2/5 red. So what is the
  ratio of green to red in the rectangle?
• Have a crack at it. How many greens do you get
  for every two reds?

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• The ratio here is 32. But if you said 1.51,
  then youre right too. Clearly for every one and
  a half greens you get one red. And for every 3
  greens you get 2 reds. And for every 6 greens you
  get 4 reds, so the ratio is 64 as well.
• You can play that game all day because you dont
  change a ratio by multiplying both sides by the
  same number. In fact its just like fractions.
  You know that
• though you would probably never put a decimal in
  a numerator like we have just done with 1.5. So
  with ratios its true that 32 64 1812
  2114 1.51.
• Ratios keep things in the same proportion. So if
  you have 3 apples for every 2 oranges, then you
  have to have 6 apples for every 4 oranges, 18
  apples for every 12 oranges, 21 apples for every
  14 oranges, and 1.5 apples for every 1 orange.
  The number of apples to oranges is always in the
  same proportion (ratio).

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• Can you think of things in everyday life that are
  always in the same ratio (or nearly always in the
  same ratio)? How about things that are not in the
  same ratio?
• Pick out the things in the same ratio from the
  list below and then add your own items
• fingers per hand
• cents per dollar
• tries per rugby game
• hours every 7 days
• dollars per litre of petrol
• tails per monkey
• income per hour
• income tax per dollar
• eyes per netball team
• height per tree.
• The items in (i), (ii), (iv), (vi), (vii) (unless
  you take into account overtime pay), and (ix) are
  always in a constant ratio.
• What list did you come up with?

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• Lets turn that around. If I give you a ratio,
  say 53, can you divide a rectangle up into that
  ratio? Have a crack at that. How will you do it?

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• OK. For something that there are 3 of, there have
  to be 5 of the other thing. If we go back to
  coloured pieces, suppose that for every 5 lots of
  green there are 3 lots of red. So in 8 coloured
  pieces, 5 are green and 3 are red. So that gives
  us something like this.

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• But thats not the only way the two colours can
  be arranged. You may well have put 8 squares in a
  row to make an 8 by 1 rectangle. Even then you
  didnt have to colour them so that all the red
  was together and all the green was together.
• Here are just three ways to colour a rectangle so
  that green to red is 53.

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• So now divide a rectangle up so that the ratio of
  its blue to green parts is 73.
• And do that in about five ways.

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• Of course there are an infinite number of
  possibilities here but at the end of the day, the
  simplest way to do this is to first notice that 7
  3 10. Then divide the rectangle up into
  tenths. The easiest way to do this is to make the
  rectangle out of 10 squares.
• From here, 7 of the squares have to be blue and 3
  have to be green.

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• Now that you have mastered that, how about
  dividing a rectangle up so that the ratio of its
  blue to red parts is 27?
• We havent had a ratio where the second part was
  bigger than the first part but that shouldnt
  worry you. Exactly the same principle applies.

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• Its getting a bit boring with the coloured
  rectangles. Lets apply ratios to fruit. Suppose
  we want to divide some fruit up so that the ratio
  of the fruit that I get to the fruit that you get
  is 27. Then we apply exactly the same principle
  to fruit as we did to area.
• First think about the fact that 2 7 9. Then
  arrange for every 9 pieces of fruit that I get 2
  and you get 7.
• Actually, when you think about it, what has
  happened here is that for every 9 pieces of fruit
  I get 2/9 and you get 7/9. So we dont even have
  to stick to whole pieces of fruit.

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• So if you give me 1/6 of a pizza and you take
  5/6, then you have split the pizza from me to you
  in the ratio of 15.
• And if youve divide 100 between me and you in
  the ratio of 91, then youve divided it up into
  tenths thats lots of 10. And I have got 9/10
  (thats very generous of you, thanks) and you
  have got 1/10. So I take away 90 and you take
  away 10.

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• Suppose that the exchange rate today for
  Wellandian dollars is 50, in other words for
  every 1 New Zealand you get 50 Wellandian, then
  we have a ratio problem to find the number of
  Wellandian dollars you can get for 3 New
  Zealand. This is because the exchange rate is a
  ratio of 150. For the first 1 New Zealand you
  get 50 Wellandian for the second 1 New Zealand
  you get 50 Wellandian and for the third 1 New
  Zealand you get 50 Wellandian. So for 3 New
  Zealand you get 150 Wellandian. You see that the
  ratio of 150 is the same as 3150. And thats
  the way to calculate how to exchange from one
  currency to the next

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• But just when you were feeling confident about
  this ratio business, the Principal comes in. Is
  it possible to divide that 100 we were talking
  about a little while ago in the ratio of 433
  for me to the Principal to you?
• But why is that hard? Think about what we did
  with the simpler ratios. 4 3 3 10. That
  suggests that we break everything up into tenths.
  So I get 4/10, the Principal gets 3/10 and you
  get 3/10 too. Ok, Im off with 40, the Principal
  gets 30 and you get 30 too.

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• How about dividing a rectangle up into red to
  white to blue in the ratio of 451?
• Or how about dividing the angles in a triangle
  into the ratio 567? (Bear in mind that there
  are 180 degrees in a triangle.)
• What fraction of the fruit do we all get if you
  divide the fruit between you, me and your brother
  in the ratio of 321?
• And if in a basketball game you get 2/9 of the
  points while I get 3/9 of the points and your
  sister gets the rest, in what ratio did we divide
  the points?

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• Please email us at derek_at_nzmaths.co.nz for any
  correspondence related to this workshop.