Breakfast in Sydney; Dinner in London

• 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.)

• 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?• (You’ll probably need a calculator for this.)

• That’s far too hard so let’s creep up on it. Let’s 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?

• That’s not so bad. Surely we could just multiply by 2. If$NZ 1 = $AUD .5 then$NZ 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.

• 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.

0 1 2 3 NZD

0 0.5 1 1.5 AUD

• But there’s another way to look at this. The ratio of Australian dollars to New Zealand dollars is fixed no matter what’s amount we’re thinking about. So the cost of breakfast in NZD/ the cost of breakfast in

AUD = 1/0.5,

• and that givesthe cost of breakfast in NZD = 1 x the cost of breakfast

in AUD/0.5.

• So we have another way to do this problem.

• Now let’s 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.

• 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.

• Now let’s 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.

• 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.

0 0.9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 9.0 AUD

0 1 2 3 4 5 6 7 8 9 10 NZD

• OK, let’s go for the big one. What if $NZ1 = $AUD0.9083? How expensive is that breakfast?

• Try it a couple of ways.

• 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 we’d have $NZ 10000 = $AUD 9083.

• So$NZ 10000/9083 = $AUD 1.

• And that comes out to $NZ 1.1009578333149840361114169327315 = $AUD 1.

• That’s not the sort of number that you’d want to remember. But if you have your calculator at hand you’ll find that breakfast is now costing $16.79.

• 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?

• 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?

• 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.

• That’s all very well but why do we not quite get the same answer by dividing that we got by multiplying?

• Where is the mistake?

• Well it’s 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.

• 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?

• Well, there’s 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?!?

• 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 you’re spending you’re going to need a calculator with you all the time or get a friend who’s 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 it’s as rough as you need it to be.

• 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 let’s approximate to 1.10. At that rate the breakfast is about $15.25 (that’s the 1) + $1.53 (and that’s the .1) = $16.78. That’s nearer the right price. Will that break the bank? And is the extra accuracy gained for a little bit more arithmetic worth it?

• 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

• The decision here is up to you. We don’t 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

• 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?

• 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 you’ll 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

• Now you know that there’s always going to be one currency that’s 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, that’s 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 that’s not too good for large values. To do better you’ll 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; that’s 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 Krone’s 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.

• Please email us at derek@nzmaths.co.nz for any correspondence related to this workshop.