Currency exchange
One dollar trades for a fixed number of euros. How do you turn any amount of one country’s money into another’s, which way round do you multiply, and when does the trade stop being that simple?
Where you are 13 in Time and money
When you reach for it
Most countries have their own money, their own currency: dollars in the United States, euros across much of Europe, pounds in Britain, yen in Japan, pesos in Mexico and in the Philippines. A traveler landing in Spain with dollars needs euros. Someone in Ohio buying a sketchbook from a shop in Tokyo sees its price in yen but pays in dollars. Someone in the United States who sends money to family in the Philippines hands over dollars, and the family collects pesos. Whenever an amount of one currency has to be stated in another, you reach for an exchange rate.
An exchange rate says how much of one currency trades for one unit of another, for example
Real rates change from day to day, and from one bank, booth or website to the next. Every rate in this lesson is an example, made up but of a realistic size; a real trade uses the rate quoted on the day. Amounts of money are decimals counted in hundredths, as in money written as decimals.
The procedure
The rate is a constant of proportionality. At 1 dollar to 0.92 euros, every dollar trades for the same 0.92 euros, whether it is the first dollar handed over or the three-hundredth, and no dollars trade for no euros. So the number of euros is proportional to the number of dollars, and the rate is the constant, 0.92 euros per dollar (the constant of proportionality):
Multiply going out, divide coming back. The rate tells how much of the second currency goes with one unit of the first. Starting from the currency on the “1” side, multiply by the rate. Starting from the other currency, divide by the rate, because dividing undoes multiplying: if \text{euros} = 0.92 \times \text{dollars}, then \text{dollars} = \text{euros} \div 0.92. Dividing by 0.92 is the same as multiplying by \displaystyle \frac{1}{0.92} \approx 1.087, which is the rate the other way round: 1 euro is about 1.087 dollars.
Let the units decide. Write the rate as a fraction with its units, the right way up, so that the unit you start with cancels:
The fraction is worth exactly one trade, 0.92 euros for 1 dollar, the same money on top and bottom, so multiplying by it changes the currency and not the value (conversion factors). To go back, turn it over: \displaystyle \frac{1\,\text{dollar}}{0.92\,\text{euros}}, and now euros cancel.
A proportion says the same thing. Put the same currency on top on both sides, the same currency underneath, and solve the proportion:
Setting the cross products equal gives e = 0.92 \times 300 = 276, the same multiplication.
Size check. Here one dollar buys fewer than one euro, so an amount written in euros is a smaller number than the same amount in dollars. At 150 yen to the dollar, an amount in yen is a much larger number than in dollars. Before using an answer, ask whether it moved the right way.
Round once, at the end, to the smallest coin. Dollars, euros, pounds and pesos are counted to the hundredth (the cent, or penny, or centavo); yen are counted in whole yen. Rounding a rate or an amount part way through can move the final answer by several cents (Rounding decimals).
A fixed fee is handled on its own. A booth or a money-sending service may charge a flat fee, taken from the money handed over or added on top. The rate applies only to the money actually exchanged, so find that amount first and then use the rate. With a fee, the total handed over is no longer proportional to what comes out, as Examples 4 to 6 show.
Worked examples
Changing dollars into euros
Mireille is flying to Lisbon and changes 300 dollars into euros at an example rate of 1 \text{ dollar} = 0.92 euros. How many euros does she get?
SolutionThe rate has dollars on its “1” side and she starts with dollars, so multiply by the rate. Every dollar brings the same 0.92 euros, so 300 dollars bring 300 times as many:
She gets 276 euros. The units agree: dollars times euros per dollar leaves euros. As a proportion, \displaystyle \frac{1}{0.92} = \frac{300}{e}, and cross products give e = 0.92 \times 300 = 276, the same answer. Size check: each dollar is worth a little less than a euro’s worth of euros, so the euro number should be a little less than 300, and it is.
Changing euros back, and rounding
At the end of the trip Mireille has 46 euros left. A currency website shows the same example rate, 1 \text{ dollar} = 0.92 euros, in both directions.
(a) How many dollars are 46 euros worth?
(b) Her friend has 85 euros left. How many dollars are they worth, to the nearest cent?
(c) The friend rounds the reverse rate to 1.09 dollars per euro first. What answer does that give?
Solution(a) Now she starts from euros, the side that is not 1, so divide by the rate. Every dollar is 0.92 euros, so the number of dollars is the number of times 0.92 goes into 46:
46 euros are worth 50 dollars. Check by going forward: 50 \times 0.92 = 46.
(b) 85 \div 0.92 = 92.3913\ldots, and dollars are counted to the cent, so 85 euros are worth about 92.39 dollars. Rounding happens here, once, at the very end.
(c) 85 \times 1.09 = 92.65 dollars, which is 26 cents more than the answer to (b). The true reverse rate is \displaystyle \frac{1}{0.92} = 1.0869\ldots, and rounding it up to 1.09 adds a little to every euro, which builds up over 85 of them. Divide by the rate as given, or keep the reverse rate to more places.
Shopping from a shop abroad
Marisol orders a sketchbook from a shop in Japan. It costs 2{,}850 yen, and shipping costs 1{,}350 yen. Her card uses an example rate of 1 dollar = 150 yen.
(a) What does the order cost in dollars? A shop near her sells the same sketchbook for 31 dollars. Which is cheaper?
(b) Last month a friend ordered from the same shop and paid 32 dollars for a 4{,}640 yen order. What rate did the card use then, and what would Marisol’s order have cost at that rate?
Solution(a) The order is 2{,}850 + 1{,}350 = 4{,}200 yen. The rate has dollars on its “1” side, and Marisol starts from yen, so divide:
The order costs 28 dollars, which is 3 dollars less than the local shop. Size check: yen numbers are much larger than dollar numbers at this rate, and 28 is much smaller than 4{,}200.
(b) One trade is a pair of matching amounts, and the rate is their quotient, just as the constant of proportionality is the quotient of any pair:
At that rate, Marisol’s order would have cost 4{,}200 \div 145 = 28.9655\ldots, about 28.97 dollars. Fewer yen to the dollar means each yen costs more dollars, so the same order cost more last month.
Two booths, and a fee
Two exchange booths at an airport show example rates.
- Booth A: 1 dollar = 0.93 euros, with a fee of 4 dollars taken from the money handed over.
- Booth B: 1 dollar = 0.90 euros, with no fee.
(a) Which booth gives more euros for 100 dollars?
(b) Which gives more for 300 dollars?
(c) Is the number of euros proportional to the dollars handed over at each booth?
Solution(a) At booth A the fee comes off first, so only 100 - 4 = 96 dollars are exchanged: 96 \times 0.93 = 89.28 euros. At booth B all 100 dollars are exchanged: 100 \times 0.90 = 90 euros. Booth B gives 0.72 euros more.
(b) Booth A: 300 - 4 = 296 dollars are exchanged, giving 296 \times 0.93 = 275.28 euros. Booth B: 300 \times 0.90 = 270 euros. Now booth A gives 5.28 euros more.
(c) At booth B every dollar brings 0.90 euros, so the euros are proportional to the dollars, with constant 0.90. At booth A, divide the euros by the dollars handed over:
The quotients differ, so booth A’s euros are not proportional to the dollars handed over. The fee is paid once, however much is changed, so it weighs more on a small trade than on a large one. The euros are still proportional to the dollars exchanged (the amount after the fee), with constant 0.93.
Sending money to family
Joaquin sends money from the United States to his grandmother in the Philippines. The sending service uses an example rate of 1 dollar = 56 Philippine pesos and adds a fee of 5 dollars on top of the amount sent.
(a) Joaquin pays 105 dollars in all. How many pesos does his grandmother collect?
(b) She needs 14{,}000 pesos for a repair. How much must Joaquin pay in all?
(c) Are the pesos collected proportional to the dollars Joaquin pays?
Solution(a) The fee is on top, so of the 105 dollars only 105 - 5 = 100 are sent. Each of those brings 56 pesos: 100 \times 56 = 5{,}600 pesos.
(b) Work backwards. The pesos come from the dollars sent, so divide by the rate first: 14{,}000 \div 56 = 250 dollars must be sent. The fee is added after that: 250 + 5 = 255 dollars in all.
(c) Pesos per dollar paid are \displaystyle \frac{5{,}600}{105} = 53.33\ldots and \displaystyle \frac{14{,}000}{255} = 54.90\ldots, which differ, so no. The pesos are proportional to the dollars sent, with constant 56; the flat fee is what breaks the proportion for the total paid. Sending 250 dollars once costs one fee, while sending it as two transfers of 125 dollars would cost two.
Where the two booths break even
Booths A and B from Example 4 swap places between 100 and 300 dollars. Find the amount at which they give the same number of euros.
SolutionCall the dollars handed over x. Booth A exchanges x - 4 dollars, so by the distributive property it gives
and booth B gives 0.90x euros. Compared with booth B, booth A pays 0.03 euros more on every dollar and loses 3.72 euros once, the fee’s worth at its rate. The two balance when the extra covers the loss:
Check: booth A changes 120 dollars, 120 \times 0.93 = 111.60 euros, and booth B gives 124 \times 0.90 = 111.60 euros. Below 124 dollars booth B is better, above it booth A, which agrees with Example 4.
From euros to pounds by way of dollars
Lina has 230 euros left after a trip to Spain and is going on to London. The booth quotes every currency against the dollar, with example rates 1 dollar = 0.92 euros and 1 dollar = 0.80 pounds. How many pounds are her euros worth, and what is the rate from euros to pounds?
SolutionThe two rates share the dollar, so go through it. Euros to dollars is a division, since euros are not on the “1” side: 230 \div 0.92 = 250 dollars. Dollars to pounds is a multiplication: 250 \times 0.80 = 200 pounds.
The units show why this works. Each fraction is worth one, and the dollars cancel between them:
The two fractions multiply to one rate from euros to pounds:
and \displaystyle 230 \times \frac{20}{23} = 200 again. Using the rounded 0.87 gives 230 \times 0.87 = 200.10 pounds, ten pence off, which is why rounding waits until the end.
Boundaries
- A rate belongs to one trade. It holds at one place, at one time. Rates change daily, and a bank, a booth, a card and a sending service may quote different rates for the same two currencies on the same day.
- A booth quotes two rates, and they are not reciprocals. A booth that gives 0.88 euros per dollar but pays only 1.06 dollars per euro turns 100 dollars into 88 euros, and those 88 euros back into 88 \times 1.06 = 93.28 dollars. Each direction is proportional on its own; the gap between the two rates is how the booth earns. Only a single rate used both ways, as on a currency website, lets you divide to undo a trade exactly.
- A fixed fee breaks the proportion between the money handed over and the money that comes out. The rate still applies to the amount actually exchanged.
- Fees charged as a percent of the amount are another kind of charge; they belong with percent in money situations.
- Rounding. Amounts are paid in whole cents (or whole yen), so a trade that goes there and back can come out a cent different even at one rate with no fee (Exercise 19).
Where this goes next
Because an exchange is proportional, its graph is a straight line through (0, 0), as in recognizing a proportional relationship in a graph, and \text{euros} = 0.92 \times \text{dollars} is an equation of the form y = kx, as in recognizing a proportional relationship in an equation. Comparing two rates with no fee is deciding which of two ratios is larger. A booth with a flat fee follows a straight line that does not pass through (0, 0), the kind of relationship studied in modeling with linear relationships, and finding where two booths break even becomes solving an equation. Chaining rates through a shared currency is the same move as chaining any conversion factors, from inches to meters or from kilometers per hour to meters per second.
Exercises
Every rate below is an example. Round dollars, euros, pounds and pesos to the cent and yen to the whole yen, at the end.
Changing money at a given rate
- At 1 dollar = 0.92 euros, how many euros do 450 dollars buy?
- At 1 dollar = 18.50 Mexican pesos, (a) change 120 dollars into pesos, and (b) change 925 pesos into dollars.
- At 1 pound = 1.25 dollars, (a) change 80 pounds into dollars, and (b) change 210 dollars into pounds.
- At 1 dollar = 150 yen, a phone case from an online shop in Japan costs 2{,}400 yen. What does it cost in dollars? How many yen can be spent with a budget of 35 dollars?
- At 1 US dollar = 1.36 Canadian dollars, (a) change 75 US dollars into Canadian dollars, and (b) change 75 Canadian dollars into US dollars.
- Dara changes 140 dollars into euros at 1 dollar = 0.92 euros, spends 95.80 euros on her trip, and changes the rest back at the same rate. How many dollars does she get back?
Finding and comparing rates
- Farid handed over 250 dollars and received 230 euros, with no fee. Find the rate in euros per dollar and in dollars per euro, to three decimal places.
- A family received 33{,}200 Indian rupees when 400 dollars were sent to them, with the fee paid separately. Find the rate, and how many rupees 150 dollars would deliver at that rate.
- The same headphones cost 13{,}500 yen from a shop in Japan and 125 Canadian dollars from a shop in Canada, shipping included. With 1 dollar = 150 yen and 1 dollar = 1.36 Canadian dollars, which is cheaper in US dollars, and by how much?
- Booth X gave a traveler 45.50 euros for 50 dollars, and booth Y gave another 73.60 euros for 80 dollars, neither charging a fee. Which booth has the better rate for someone changing dollars into euros, and how many more euros would it give for 300 dollars?
Fees and two-way rates
- A booth gives 0.90 euros per dollar and takes a 5 dollar fee from the money handed over. How many euros come out for 105 dollars, and for 205 dollars? Is the number of euros proportional to the dollars handed over?
- A sending service uses 1 dollar = 129 Kenyan shillings and adds a fee of 3 dollars on top. (a) How much must be paid in all for a family to collect 25{,}800 shillings? (b) How many shillings are collected when 53 dollars are paid in all?
- Booth P gives 18.20 Mexican pesos per dollar with no fee. Booth Q gives 18.60 pesos per dollar and takes a 6 dollar fee from the money handed over. (a) Which gives more pesos for 200 dollars? (b) For 500 dollars? (c) For what amount do they give the same?
- A booth sells pounds at 0.76 pounds per dollar and buys them back at 1.24 dollars per pound. Change 250 dollars into pounds and straight back. How much is lost, and what would the buy-back rate have to be, to three decimal places, for nothing to be lost?
Chains of rates
- With 1 dollar = 0.92 euros and 1 dollar = 150 yen, change 69 euros into yen by way of dollars. Find also the rate in yen per euro, to the nearest hundredth.
- With 1 US dollar = 1.36 Canadian dollars and 1 US dollar = 18.50 Mexican pesos, how many pesos are 340 Canadian dollars worth?
Reasoning
- At 1 dollar = 0.92 euros, explain why you multiply by 0.92 to change dollars into euros but divide by 0.92 to change euros into dollars, and why 276 euros and 300 dollars can be the same amount of money although 276 is the smaller number.
- A currency website prints 1 dollar = 0.92 euros and, below it, 1 euro = 1.09 dollars. Show that 0.92 \times 1.09 is not exactly 1, and explain how both lines can still be correct.
- Using 1 dollar = 0.92 euros in both directions and rounding to the cent each time, change 10.19 dollars into euros and then change those euros back. Explain why the result is not 10.19 dollars, though no fee was paid.
- Explain, using units, why the rate in pounds per euro is the rate in pounds per dollar divided by the rate in euros per dollar.
- With a flat fee taken from the money handed over, explain why the money that comes out is not proportional to the money handed over, though it is proportional to the money exchanged. What comes out when exactly the fee is handed over?
Answers
- 450 \times 0.92 = 414 euros. Dollars are on the “1” side, so multiply.
- (a) 120 \times 18.50 = 2{,}220 pesos. (b) 925 \div 18.50 = 50 dollars.
- (a) 80 \times 1.25 = 100 dollars, since pounds are on the “1” side. (b) 210 \div 1.25 = 168 pounds.
- 2{,}400 \div 150 = 16 dollars, and 35 \times 150 = 5{,}250 yen.
- (a) 75 \times 1.36 = 102 Canadian dollars. (b) 75 \div 1.36 = 55.147\ldots, about 55.15 US dollars.
- She gets 140 \times 0.92 = 128.80 euros and has 128.80 - 95.80 = 33 euros left. Back in dollars: 33 \div 0.92 = 35.869\ldots, about 35.87 dollars.
- 230 \div 250 = 0.92 euros per dollar, and 250 \div 230 = 1.0869\ldots, about 1.087 dollars per euro. The two rates are reciprocals.
- 33{,}200 \div 400 = 83 rupees per dollar, so 150 dollars deliver 150 \times 83 = 12{,}450 rupees.
- Japan: 13{,}500 \div 150 = 90 dollars. Canada: 125 \div 1.36 = 91.911\ldots, about 91.91 dollars. The shop in Japan is cheaper by about 1.91 dollars.
- Booth X: 45.50 \div 50 = 0.91 euros per dollar. Booth Y: 73.60 \div 80 = 0.92 euros per dollar. Booth Y is better: 300 \times 0.92 = 276 euros against 300 \times 0.91 = 273, so 3 euros more.
- (105 - 5) \times 0.90 = 90 euros and (205 - 5) \times 0.90 = 180 euros. Not proportional: \displaystyle \frac{90}{105} \approx 0.857 but \displaystyle \frac{180}{205} \approx 0.878. Doubling the euros took less than double the dollars, because the fee is paid once.
- (a) 25{,}800 \div 129 = 200 dollars must be sent, so 200 + 3 = 203 dollars in all. (b) 53 - 3 = 50 dollars are sent, so 50 \times 129 = 6{,}450 shillings.
- (a) P: 200 \times 18.20 = 3{,}640 pesos. Q: 194 \times 18.60 = 3{,}608.40 pesos. P gives more. (b) P: 500 \times 18.20 = 9{,}100 pesos. Q: 494 \times 18.60 = 9{,}188.40 pesos. Q gives more. (c) Q gives 18.60(x - 6) = 18.60x - 111.60 and P gives 18.20x, so they are equal when 0.40x = 111.60, that is x = 279 dollars. Check: both give 5{,}077.80 pesos.
- 250 \times 0.76 = 190 pounds, then 190 \times 1.24 = 235.60 dollars, so 250 - 235.60 = 14.40 dollars are lost. Nothing would be lost at the reciprocal rate, 1 \div 0.76 = 1.3157\ldots, about 1.316 dollars per pound.
- 69 \div 0.92 = 75 dollars, and 75 \times 150 = 11{,}250 yen. The rate is 150 \div 0.92 = 163.043\ldots, about 163.04 yen per euro.
- 340 \div 1.36 = 250 US dollars, and 250 \times 18.50 = 4{,}625 pesos.
- Each dollar trades for 0.92 euros, so d dollars trade for d lots of 0.92, which is 0.92 \times d. Going back asks how many dollars make up a given number of euros, that is, how many times 0.92 fits into it, which is a division; dividing undoes the multiplication. A euro is a bigger unit than a dollar at this rate (one euro is worth more than one dollar), so the same money needs fewer euros than dollars, just as a length needs fewer meters than centimeters.
- 0.92 \times 1.09 = 1.0028, not 1. The exact reverse rate is 1 \div 0.92 = 1.0869\ldots, which rounds to 1.09. Both printed rates are the same rate, each rounded to two decimal places, so their product is close to 1 but not exactly 1.
- 10.19 \times 0.92 = 9.3748, paid as 9.37 euros. Then 9.37 \div 0.92 = 10.1847\ldots, paid as 10.18 dollars, one cent less. Rounding to the cent dropped 0.0048 euros on the way out, and that small piece is worth about half a cent, enough to tip the second rounding down.
- Pounds per dollar is \displaystyle \frac{\text{pounds}}{\text{dollar}} and euros per dollar is \displaystyle \frac{\text{euros}}{\text{dollar}}. Dividing by a fraction is multiplying by its reciprocal, so the quotient is \displaystyle \frac{\text{pounds}}{\text{dollar}} \times \frac{\text{dollar}}{\text{euros}}, and the dollars cancel, leaving pounds per euro. In numbers, one euro is \displaystyle \frac{1}{0.92} dollars, and each of those dollars is 0.80 pounds, so one euro is \displaystyle \frac{0.80}{0.92} pounds.
- The money that comes out is the rate times the money exchanged, and the money exchanged is the money handed over minus the fee, so the rate applies to the exchanged amount with one constant. For the money handed over there is no single constant: handing over exactly the fee leaves nothing to exchange, so that amount brings out 0, while a proportional relationship pairs only 0 with 0. The quotient of what comes out by what is handed over grows as the amount grows, because the fee is shared over more dollars.