Changing money
Tonight's idea
An exchange rate is the price of one kind of money in another. Every rate on this sheet is an invented round number. Real rates change daily.
Say 1 dollar buys 0.90 euros. That is an amount "per one", so to change dollars into euros, multiply by 0.90: $200 becomes \(200 \times 0.90 = 180\) euros. To go back, undo it: \(180 \div 0.90 = 200\) dollars. There and back, nothing is lost.
A money-changing booth does not work for nothing. It shows two rates, a little apart. It gives you fewer euros for each dollar than the fair rate. When you come back, it asks more euros for each dollar.
Fair rate 0.90. The booth pays 0.88 euros for each dollar you hand in, and asks 0.92 euros for each dollar you want back. The gap is called the spread.
The spread is the booth's pay. You never see a fee. You just get a little less each way.
Watch one: $230 into euros and straight back
Out: the booth gives 0.88 euros per dollar. \(230 \times 0.88 = 202.40\) euros.
Back: it wants 0.92 euros for each dollar. \(202.40 \div 0.92 = 220\) dollars.
You started with $230 and now hold $220. The booth kept $10 and you bought nothing. At the fair rate both ways you would have all $230 back.
Your turn
- Use the fair rate, 1 dollar = 0.90 euros. (a) Change $50 into euros. (b) Change 63 euros into dollars. (c) Another invented rate: 1 dollar = 150 yen. Change 12,000 yen into dollars.
- Headphones cost 54 euros in Paris and $62 at home. At the fair rate of 0.90, which is cheaper, and by how many dollars?
- An airport booth pays 0.80 euros per dollar and asks 1.00 euro for each dollar back. (a) Change $300 into euros. (b) Change it all back. (c) What percent of your money did the booth keep?
- Suppose 1 dollar = 0.80 euros. (a) How many dollars is 1 euro worth? Leo says $1.20, "because a dollar is 20 cents short of a euro". Show that he is wrong. (b) If 1 euro = 150 yen as well, how many yen is 1 dollar worth?
Stretch: round and round
At the airport booth, each trip there and back leaves you 80% of what you had. Start with $1,000 and keep going round. After how many round trips do you first have less than half? Why is "five trips lose 100%" wrong?
Answers
- (a) \(50 \times 0.90 = 45\) euros. (b) \(63 \div 0.90 = 70\) dollars. (c) \(12{,}000 \div 150 = 80\) dollars.
- \(54 \div 0.90 = 60\) dollars. Paris is cheaper, by $2.
- (a) \(300 \times 0.80 = 240\) euros. (b) \(240 \div 1.00 = 240\) dollars. (c) It kept $60 out of $300, which is 20%.
- (a) \(1 \div 0.80 = 1.25\) dollars. Check: \(1.25 \times 0.80 = 1\) euro exactly. Leo's $1.20 buys only \(1.20 \times 0.80 = 0.96\) euros. The 20 cents he noticed are euro cents, a fifth of a euro; the gap seen from the dollar side is a quarter of a dollar. (b) \(0.80 \times 150 = 120\) yen.
Stretch. $1,000 becomes $800, then $640, then $512, then $409.60. So 4 round trips. Each 20% is taken from what is left, which is smaller every time: $200, then $160, then $128. The losses shrink, so they never add up to everything.