The price of one unit of money
Each country has its own money, called its currency. The United States uses dollars and much of Europe uses euros. Before a trip, a traveler changes some dollars into euros at a bank.
The exchange rate says how much of one currency one unit of the other buys. Real exchange rates change from day to day, so this lesson assumes a rate of $1 = 1.5 euros, and a question always states the rate it assumes. Every dollar buys the same 1.5 euros, so this is a rate, just like 12 liters per hour: 1.5 euros per dollar.
Each dollar on the top line is paired with 1.5 euros on the bottom line. The mark under $1 is the exchange rate.
Dollars into euros: multiply
How many euros does $4 buy? Each of the 4 dollars buys 1.5 euros, so multiply: 4 × 1.5 = 6 euros.
At this rate one dollar is worth more than one euro, so the same money is always a bigger number of euros than of dollars. $4 is 6 euros, and 6 is more than 4, which is a quick check that the answer went the right way.
4 steps of 1.5 euros: $4 sits above 4 × 1.5 = 6 euros.
Euros back into dollars: divide
Now go the other way. How many dollars are 6 euros worth? Every 1.5 euros is one dollar, so count how many lots of 1.5 euros there are in 6 euros: 6 ÷ 1.5 = 4. The 6 euros are worth $4.
Changing back divides by the same rate that changing forward multiplied by. Dividing undoes multiplying, so you arrive back at the $4 you started with. Check: 4 × 1.5 = 6.
The same pairs read from euros to dollars: 6 euros is 4 lots of 1.5 euros, so it sits above $4.
Which way does the rate point?
The rate $1 = 1.5 euros tells you what one dollar buys. When you have dollars, each dollar buys 1.5 euros, so multiply the number of dollars by 1.5. When you have euros, find how many lots of 1.5 euros they make, which is a division.
Some rates are less than 1. If $1 = 0.8 euros, a dollar is worth less than a euro. Then $20 buys 20 × 0.8 = 16 euros, and 16 euros are worth 16 ÷ 0.8 = $20. The rule does not change: from dollars to euros, multiply by the number of euros one dollar buys; from euros to dollars, divide by it.
The usual mistakes
Multiplying both ways. Changing 6 euros into dollars by working out 6 × 1.5 = 9 gives $9, more dollars than the $4 that bought those euros. Coming back divides: 6 ÷ 1.5 = $4.
Keeping the number. $6 is not 6 euros. A euro and a dollar are different amounts of money, so the amount must be changed at the rate.
Dividing toward euros. $6 at $1 = 1.5 euros is 6 × 1.5 = 9 euros. Dividing gives 4 euros, which is the way back to dollars.
Two different rates
A bank sells a currency at one rate and buys it back at a slightly different rate, so a traveler who changes money and then changes it back usually ends up with a little less. In the next problem, each direction has its own rate, and each rate is used only in its own direction.
Worked example: Dollars Changed into Euros for a Trip, and the Leftover Euros Changed Back
Question Before a trip to Italy, Lena changed $800 into euros at the rate $1 = €0.90. She spent €630 on the trip. When she came home, her bank bought back all her leftover euros at the rate €1 = $1.05. (a) How many euros did Lena have left at the end of the trip? (b) How many dollars did the bank pay her for them, and how many dollars less was that than those euros had cost her before the trip?
1.Cut $800 into 8 units of $100. At $1 = €0.90, each unit buys 100 × 0.90 = 90 euros.
Cut $800 into 8 units of $100; each unit buys 100 × 0.90 = 90 euros. 2.So Lena received 8 × 90 = 720 euros.
Lena received 8 × 90 = 720 euros. 3.She spent €630, which is 630 ÷ 90 = 7 units, so 1 unit is left.
She spent 630 euros, which is 7 units, so 1 unit is left. 4.(a) €90 were left.
(a) €90 were left. 5.The bank pays 90 × 1.05 = $94.50 for that unit.
The bank pays 90 × 1.05 = $94.50 for them. 6.The same unit of euros had cost her $100 before the trip.
That unit of euros had cost her $100 before the trip. 7.(b) The bank paid $94.50, which is 100 − 94.50 = $5.50 less. Check: 94.50 + 5.50 = 100.
(b) The bank paid 100 − 94.50 = $5.50 less than the euros had cost.
Answer: (a) €90; (b) $94.50, which is $5.50 less than the $100 those euros had cost
Common mistakes
- Changing the €90 back with the first rate, to get 90 ÷ 0.90 = $100: the bank uses its own rate for the way back, €1 = $1.05.
- Multiplying €90 by 0.90 to change it back: 0.90 is the number of euros for each dollar, so it changes dollars into euros, never euros into dollars.