To buy something from another country, you usually need its currency. An exchange rate is just a price, and converting between currencies is a proportion.
Key idea
An exchange rate is the price of one currency in terms of another, such as 0.90 euros per dollar. Always read the units. If the number of euros per dollar rises, each dollar buys more euros: the dollar has appreciated and the euro has depreciated. The two ways of quoting are reciprocals: 0.90 euros per dollar is about 1.11 dollars per euro.
Why it works
"Euros per dollar" is the price of a dollar, in euros. When that price goes up, the dollar is worth more: it appreciated. The same move, seen from the other side, makes the euro cheaper in dollars. Converting is unit cancelling: $\$200 \times 0.90 \dfrac{\text{euros}}{\text{dollar}} = 180$ euros. Mixing up which currency is "per" is the most common error, so write the units every time.
Worked examples
Worked example: Converting
The rate is 150 yen per dollar. How many yen do you get for \$40? How many dollars for 3,000 yen?
- $40 \times 150 = 6{,}000$ yen.
- $3{,}000 \div 150 = 20$ dollars.
Answer: 6,000 yen; \$20
Worked example: Which currency appreciated?
The rate moves from 140 to 150 yen per dollar.
- A dollar now buys more yen: the dollar appreciated.
- A yen now buys fewer dollars ($\dfrac{1}{150}$ instead of $\dfrac{1}{140}$): the yen depreciated.
Answer: Dollar up, yen down
Worked example: Percent changes
The rate goes from 100 to 125 yen per dollar. By what percent did each currency change?
- Dollar: $\dfrac{125 - 100}{100} \times 100 = 25\%$ appreciation.
- Yen: from $\dfrac{1}{100}$ to $\dfrac{1}{125}$ dollars, a change of $\dfrac{100}{125} - 1 = -20\%$, a 20% depreciation. The two changes are not the same size.
Answer: Dollar +25%, yen −20%
Common mistake
Reading a rise in yen per dollar as the yen appreciating. It means the dollar has appreciated and the yen has depreciated. Check: ask "how much does one dollar buy now?" More means a stronger dollar.
See live rates for the major currency pairs on the Forex & Bitcoin page; each quote names its units.