Breakfast in Sydney; Dinner in London

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Breakfast in Sydney Dinner in London
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• The purpose of this workshop is to practice
  solving practical problems involving proportions.
  We use the context of exchange rates as an
  example. Another example of proportions in every
  day life that you could explore is converting
  between metric and imperial units. Both of these
  might fit into series of lessons based on travel.
  (See our two Level 5 units Playing With Money.)

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• When I went from NZ to Australia the exchange
  rate was NZ1 AUD0.9083. Breakfast in Sydney
  cost me AUD15.25.
• When I went on to England the exchange rate at
  Sydney airport was AUD1 0.4102. At Heathrow I
  had a meal that cost me 56.23.
• How much were the two meals in real money?
• (Youll probably need a calculator for this.)

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• Thats far too hard so lets creep up on it.
  Lets take it a step at a time but make life
  easier by taking simpler numbers to get the feel
  of what to do.
• Suppose that NZ 1 AUD .5 what is AUD1 worth
  in NZ dollars? How would you do that?
• Thats not so bad. Surely we could just multiply
  by 2. IfNZ 1 AUD .5 thenNZ 2 AUD 1.
• With a simple exchange rate like NZ 1 AUD .5,
  all you have to do is to remember to take the
  local Australian price and multiply by 2. So a
  breakfast in Sydney of 15.25 must translate to
  15.25 x 2 30.50 in New Zealand dollars.

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• We can also look at this using a double number
  line.
• Here, if we make the line long enough, we can
  read the breakfast cost straight from the line.

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• But theres another way to look at this. The
  ratio of Australian dollars to New Zealand
  dollars is fixed no matter whats amount were
  thinking about. So
• the cost of breakfast in NZD/ the cost of
  breakfast in AUD 1/0.5,
• and that gives
• the cost of breakfast in NZD 1 x the cost of
  breakfast in AUD/0.5.
• So we have another way to do this problem.

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• Now lets make it a bit more difficult. Suppose
  now that the exchange rate was NZ 1 AUD 0.75.
  The simple trick before was to multiply both
  sides by 2 to get a 1 on the Australian side.
  How can we convert 0.75 to 1?
• What is 15.25 in New Zealand dollars now?
• Do it another way to check your answer.

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• You might have found a suitable multiple or you
  might have done it geometrically using a double
  number line.
• If you double 0.75 you get 1.50. If you double
  1.50 you get 3. So we can go from 0.75 to 3 by
  multiplying by 4. That gives
• NZ 4 AUD 3.
• To get AUD1 we divide by 3, so
• NZ 4/3 AUD 1 or NZ 1.33 AUD 1.
• So our Sydney breakfast now costs NZ15.25 x 1.33
  20.2825 (if you use a calculator). That rounds
  to 20.28.

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• Now lets make it a bit more difficult. Suppose
  now that the exchange rate was NZ 1 AUD .9.
  The simple trick before was to multiply both
  sides by 2 to get a 1 on the Australian side.
  How can we convert .9 to 1?
• What is 15.25 in New Zealand dollars now?
• Do it another way to check your answer.

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• Suppose that we used a double number line and
  looked at what happened every 0.9 dollars along
  the bottom of the number line.
• From that we can see that AUD1 NZD 10/9
  NZD 1.11. (Actually the 1s go on for ever.)
• So our breakfast actually cost 15.25 x 1.11
  16.9275 16.93 (after rounding) in New Zealand
  dollars. We can check that by extending the
  number line to past 15.

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• OK, lets go for the big one. What if NZ1
  AUD0.9083? How expensive is that breakfast?
• Try it a couple of ways.

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• Of course the problem is getting a whole number
  of dollars on the Australian side. If we use the
  double number line then we might have to go to
  9083 before we get a whole number of dollars. At
  that point wed have
• NZ 10000 AUD 9083.
• So
• NZ 10000/9083 AUD 1.
• And that comes out to
• NZ 1.1009578333149840361114169327315 AUD 1.
• Thats not the sort of number that youd want to
  remember. But if you have your calculator at hand
  youll find that breakfast is now costing 16.79.

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• Hang on now. Do you notice anything here?
• If NZ 1 AUD 0.5 then NZ 2 AUD 1.
• If NZ 1 AUD 0.75 then NZ 4/3 AUD 1.
• If NZ 1 AUD 0.9 then NZ 10/9 AUD 1.
• If NZ 1 AUD 0.9083 then NZ 10000/9083
  AUD 1.
• Can we write 2 in terms of 0.5?
• Can we write 4/3 in terms of 0.75?
• Can we write 10/9 in terms of 0.9?
• Can we write 10000/9083 in terms of 0.9083?

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• If NZ 1 AUD 0.9102, then what is AUD 1?
• What can we do to 0.9102 to get 1?
• So what is another way of thinking about the NZ
  dollar equivalent of AUD 1?
• And what is another way of looking at the NZ
  dollar equivalent of AUD 15.25?

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• The secret is in division.
• 2 1/0.5, so here AUD15.25 becomes NZ
  15.25/0.5 NZ 10.25.
• 4/3 1/0.75, so here AUD15.25 becomes NZ
  15.25/0.75 NZ 20.33.
• 10/9 1/0.9, so here AUD15.25 becomes NZ
  15.25/0.9 NZ 16.94.
• 10000/9083 1/.9083, so here AUD15.25 becomes
  NZ 15.25/0.9083 NZ 16.79.

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• Thats all very well but why do we not quite get
  the same answer by dividing that we got by
  multiplying?
• Where is the mistake?

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• Well its not so much a mistake as an error.
• 1.33 is not quite 1/0.75. More accurately 1/0.75
  1.333333333333333333333333333 and then some.
  But these extra 3s make all the difference.
• In the same way 1/0.9 is more like
  1.1111111111111111111111111111111 and then some
  rather than 1.11.
• So because of that, the division is more accurate
  than the multiplication.

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• But now we are in London. NZ 1 AUD 0.9083 and
  AUD 1 0.4102. So what is the cost in New
  Zealand dollars of the 56.23 evening meal?

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• Well, theres more than one way to do this. What
  did you come up with?
• How about
• NZ 1/0.9083 AUD 1 0.4102,
• so
• NZ (1/0.9083)/(0.4012) 1,
• so
• 56.23 NZ (56.23/0.9083)/(0.4012) NZ
  154.30.
• What an expensive meal?!?

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• The problem with exchange rates is that they are
  generally not very nice numbers. That is to say
  they are usually not expressed in small whole
  numbers. So if you want to have some idea of how
  much youre spending youre going to need a
  calculator with you all the time or get a friend
  whos a mathematician.
• Since both of those are a bit extreme, and all
  the good mathematicians have gone anyway, taking
  rough approximations is probably the way to go.
  And how rough is an approximation? Well its as
  rough as you need it to be.

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• Suppose you are in Oz and you know that the
  exchange rate is that horrible NZ 1 AUD
  0.9083. Well 0.9083 is about 1, so why not assume
  that NZ 1 AUD 1. That will make your
  breakfast of AUD 15.25 cost 15.25. Of course
  you now it will be a bit more than that in Kiwi
  dollars but are you happy to pay that for
  breakfast? If you pay that for breakfast for
  every day of your 7 day stay in Sydney is that
  going to break the bank?
• On the other hand we know that 1/.9 is 1.1111. So
  lets approximate to 1.10. At that rate the
  breakfast is about 15.25 (thats the 1) 1.53
  (and thats the .1) 16.78. Thats nearer the
  right price. Will that break the bank? And is the
  extra accuracy gained for a little bit more
  arithmetic worth it?

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• OK so here are a few things that you might want
  to buy in Sydney. Try a couple of approximations
  for each. What is the rough cost of each? Can you
  afford the goods? Will you buy or go elsewhere?
  Are they a bargain?
• 1 pair of your favourite brand of Jeans AUD
  163.99
• The hire of a medium sized car for a day AUD 45
• A post card of Sydney Harbour AUD 2.98
• A 2 hour cruise of the Harbour AUD 75
• Dinner in a good restaurant at The Rocks AUD 83
  per person
• A reasonable hotel in Darling Harbour AUD 134
  per room per night

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• The decision here is up to you. We dont know
  your budget. Only you can tell if these things
  are expensive or not. But the trick is going to
  be to try to find a simple calculation that will
  give you a good idea of the Kiwi equivalent
  price.
• What are you going to do in England though? Here
  we know that NZ 1 AUD 0.9083 and AUD 1
  0.4102. What roughly is 1 in NZD? So roughly
  what is the equivalent Kiwi price of the
  following?
• 1 pair of your favourite brand of Jeans 68.99
• The hire of a medium sized car for a day 41.75
• A post card of Tower Bridge 2.60
• A 2 hour cruise of the Thames 59
• Dinner in a good restaurant near Piccadilly
  Circus 72 per person
• A reasonable hotel near Hyde Park 121 per room
  per night

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• But French Polynesia is a different story. There
  the exchange rate is NZ 1 CFP 76.80. What do
  you do to make sure that you have some control
  over your money there?

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• You can keep working on this. If you search on
  the net for Exchange Rates you can play this game
  for any country you like. But it might be fun to
  see what simple calculation youll use to find
  the approximate New Zealand prices in the
  following countries
• NZ 1 0.8621 Canadian dollars
• NZ 1 0.5733 Euros
• NZ 1 5.527 Hong Kong dollars
• NZ 1 79.23 Japanese yen
• NZ 1 4.482 Norwegian krone
• NZ 1 28.46 Thai baht

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• Now you know that theres always going to be one
  currency thats a pain. But just as a check to
  what you did
• Try CFP 75 NZ 1, so CFP 1 is about NZ 1/75
  NZ 0.013333. Hmm, thats not so easy. How
  accurate would just dividing by 100 be? That way
  all you have to do is to move the decimal point
  two places to the left.
• But thats not too good for large values. To do
  better youll then need to add on a third.
• NZ 1 0.8621 Canadian dollars, so Can 1 NZ
  1.16. Can you manage multiplying by 1.2 in your
  head?
• NZ 1 0.5733 Euros, 1 Euro NZ 1.744, which
  is about 1.75. Multiplying by 4 and dividing by 7
  will be pretty accurate by you may be satisfied
  by dividing by 2. How bad would that be? Could
  you think of a way to adjust the approximation
  for big sums?
• NZ 1 5.527 Hong Kong dollars thats much
  closer to a division by 2.
• NZ 1 79.23 Japanese yen, so 1 yen is roughly
  NZ 1/80. Can you divide by 80 easily? Move the
  decimal point and do three halvings!
• NZ 1 4.482 Norwegian krone and 1 Krones about
  NZ 0.223. Dividing by 4.5 is pretty good but
  maybe hard to do. So divide by 4 and divide by 5
  and average the answers.
• NZ 1 28.46 Thai baht gives you 30 to divide
  by. Just move the decimal point and divide by 3.

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