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Exchange rates

Ratio, rates and money · level 5, lesson 5 of 7

A question to start

Before a trip to the United States, Noor changes 100 Canadian dollars into US dollars at an airport exchange booth. Then the trip is cancelled. She walks straight back to the same booth and changes all her US dollars back into Canadian dollars.

She gets back about 93 Canadian dollars. Nobody cheated her, and no money fell out of her pocket. Where did the other 7 dollars go?

The answer is in how exchange booths set their prices, and it is all ratio.

The idea

Money from different countries

Each country (or group of countries) has its own money, called its currency. Each currency has a three-letter code used by banks:

Code Currency
CAD Canadian dollar
USD US dollar
EUR euro (used in many European countries)
GBP British pound
MXN Mexican peso

An exchange rate is a rate

An exchange rate says how much of one currency you get for one unit of another. For example:

\[1 \text{ CAD} = 0.73 \text{ USD}\]

Important: exchange rates change every day, even every minute, as people buy and sell currencies. The rates in this lesson are examples chosen to make the arithmetic clear; for a real trip, check the current rate.

An exchange rate is a unit rate (level 3): 0.73 US dollars per Canadian dollar. So converting is a direct proportion: twice as many Canadian dollars buy twice as many US dollars. Everything from lesson 1 works here.

Converting both ways

With \(1 \text{ CAD} = 0.73 \text{ USD}\):

  • CAD to USD: multiply by 0.73. 250 CAD is \(250 \times 0.73 = 182.50\) USD.
  • USD to CAD: divide by 0.73. 60 USD is \(60 \div 0.73 \approx 82.19\) CAD.

How do you remember which is which? Use a sense check: one Canadian dollar buys less than one US dollar here (0.73 is less than 1), so a Canadian amount turns into a smaller US number. If your answer goes the wrong way, you multiplied when you should have divided.

A neat way to never get it wrong is to write the rate as a fraction with units, and choose the version that cancels the unit you start with:

\[250 \text{ CAD} \times \frac{0.73 \text{ USD}}{1 \text{ CAD}} = 182.50 \text{ USD}\]
\[60 \text{ USD} \times \frac{1 \text{ CAD}}{0.73 \text{ USD}} \approx 82.19 \text{ CAD}\]

The CAD on top cancels the CAD on the bottom, like cancelling a number in a fraction. This trick is called unit analysis, and scientists use it for every kind of unit change.

The inverse rate

If \(1 \text{ CAD} = 0.73 \text{ USD}\), then dividing both sides by 0.73 gives

\[1 \text{ USD} = \frac{1}{0.73} \text{ CAD} \approx 1.37 \text{ CAD}.\]

The two rates are reciprocals of each other: multiply them and you get 1 (\(0.73 \times 1.37 \approx 1\)). Banks and newspapers may quote either one, so always read which way round it is.

Buying rate and selling rate: where Noor's money went

A bank or exchange booth is a shop that sells money. Like any shop, it buys low and sells high. So it shows two rates for each currency. Here is an example board (in Canadian dollars for each US dollar):

CAD per 1 USD
We sell USD (you buy US dollars) 1.42
We buy USD (you sell US dollars back) 1.32

The true rate in the middle, about 1.37, is called the mid-market rate. The booth sells US dollars to you above it and buys them from you below it. The gap between the two rates is called the spread, and it is how the booth earns its money (some also charge a fee on top).

Now follow Noor:

  1. She buys US dollars. Each costs her 1.42 CAD, so 100 CAD buys \(100 \div 1.42 \approx 70.42\) USD.
  2. She sells them back. Each one gets her only 1.32 CAD: \(70.42 \times 1.32 \approx 92.96\) CAD.

She lost about 7.04 dollars, about 7% of her money, to the spread. Notice that the loss does not depend on whether the dollar went up or down: it comes from the two different rates.

100 CAD 70.42 USD 92.96 CAD ÷ 1.42 × 1.32 buy high, sell low: about 7% gone

The overall effect of the round trip is to multiply by \(\dfrac{1.32}{1.42} \approx 0.93\), the ratio of the two rates. That is why the wider the spread, the more you lose. Airport booths often have wider spreads than banks, and many credit cards charge a foreign transaction fee (often around 2.5% in Canada; check your card) on purchases in another currency.

Comparing prices in different currencies

To compare two prices, convert both into the same currency first. It is the same idea as comparing unit prices: you can only compare like with like.

Worked examples

1. CAD to USD. Using \(1 \text{ CAD} = 0.73 \text{ USD}\), change 250 CAD to US dollars.

  • \(250 \times 0.73 =\) 182.50 USD. Sense check: fewer US dollars than Canadian. ✓

2. USD to CAD. Change 60 USD to Canadian dollars at the same rate.

  • \(60 \div 0.73 \approx\) 82.19 CAD. Sense check: more Canadian dollars than US. ✓

3. The inverse rate. If \(1 \text{ EUR} = 1.50 \text{ CAD}\), how many euros is 1 CAD?

  • \(1 \div 1.50 \approx 0.667\), so \(1 \text{ CAD} \approx 0.67 \text{ EUR}\).
  • A museum pass of 45 EUR costs \(45 \times 1.50 =\) 67.50 CAD.

4. Where is it cheaper? A pair of shoes costs 120 USD in a US shop and 160 CAD in a Canadian shop. Use \(1 \text{ USD} = 1.37 \text{ CAD}\).

  • Convert the US price: \(120 \times 1.37 = 164.40\) CAD.
  • The Canadian shop is cheaper by \(164.40 - 160 =\) 4.40 CAD, before any taxes or shipping.

5. Buying for a trip. At the booth above, how many Canadian dollars does it cost to buy 200 USD? After the trip you have 50 USD left. How many Canadian dollars do you get for them?

  • Buying: the booth sells at 1.42, so \(200 \times 1.42 =\) 284 CAD.
  • Selling back: the booth buys at 1.32, so \(50 \times 1.32 =\) 66 CAD.

Try it

  1. Using \(1 \text{ CAD} = 0.73 \text{ USD}\), change 300 CAD into US dollars.
  2. Using \(1 \text{ USD} = 1.37 \text{ CAD}\), change 80 USD into Canadian dollars.
  3. If \(1 \text{ EUR} = 1.50 \text{ CAD}\), how many euros do you get for 1 CAD? For 90 CAD?
  4. A video game costs 70 USD in the US and 89.99 CAD in Canada. Using \(1 \text{ USD} = 1.37 \text{ CAD}\), where is it cheaper, and by how much?
  5. A hotel costs 150 EUR a night for 3 nights. You pay with a card that converts at \(1 \text{ EUR} = 1.50 \text{ CAD}\) and adds a 2.5% foreign transaction fee. How much is that in Canadian dollars?
  6. \(1 \text{ USD} = 1.37 \text{ CAD}\) and \(1 \text{ EUR} = 1.50 \text{ CAD}\). How many US dollars is 1 euro worth? (Hint: go through Canadian dollars.)
  7. A booth sells US dollars at 1.42 CAD each and buys them at 1.32 CAD each. What percent of your money do you lose on an immediate round trip, starting from any amount?
  8. Explain why "multiply by the rate" and "divide by the rate" are both correct, but for different directions.
Answers
  1. \(300 \times 0.73 = 219\) USD.
  2. \(80 \times 1.37 = 109.60\) CAD.
  3. \(1 \div 1.50 \approx 0.67\) EUR for 1 CAD. For 90 CAD: \(90 \div 1.50 = 60\) EUR.
  4. \(70 \times 1.37 = 95.90\) CAD. Canada is cheaper by \(95.90 - 89.99 = 5.91\) CAD.
  5. \(3 \times 150 = 450\) EUR, which is \(450 \times 1.50 = 675\) CAD. The fee multiplies by 1.025: \(675 \times 1.025 = 691.875\), so about 691.88 CAD.
  6. 1 EUR is 1.50 CAD, and each CAD is \(1 \div 1.37\) USD. So 1 EUR is \(1.50 \div 1.37 \approx 1.09\) USD.
  7. Any amount is divided by 1.42 then multiplied by 1.32, so it is multiplied by \(\dfrac{1.32}{1.42} \approx 0.930\). You keep about 93.0% and lose about 7.0%, whatever the starting amount.
  8. The rate \(1 \text{ CAD} = 0.73 \text{ USD}\) says each Canadian dollar becomes 0.73 US dollars, so going CAD to USD you multiply by 0.73 (0.73 per one, times how many). Going USD to CAD you ask "how many groups of 0.73 USD are there?", which is division. Writing the rate as a fraction with units and cancelling makes the choice automatic.

Common mistake

Multiplying when you should divide. Always do a sense check: which currency is worth more per unit? If 1 CAD buys only 0.73 USD, a Canadian amount must turn into a smaller US number.

Ignoring the spread and the fees. The rate on the news is the mid-market rate. A booth or card gives you a worse rate and may add a fee. Compare the amount you actually receive, not the advertised rate.

Where it leads