Section 1
Exchange rates — what they mean
An exchange rate tells you how much of one currency you get for one unit of another.
For example: €$1 =$ £$0.87$.
This means "one euro is worth $0.87$ pounds".
This means "one euro is worth $0.87$ pounds".
Base currency: the one with $1$ beside it — the one we're quoting in.
In €$1 =$ £$0.87$, the base is the euro.
In €$1 =$ £$0.87$, the base is the euro.
Always check which currency is the base before you start. Misreading the direction of the rate is the most common error in FX problems.
Section 2
Converting from the base currency (multiply)
When you're going from the base currency to the other, multiply by the rate.
From base → multiply:
If €$1 =$ £$r$, then €$X =$ £$(X \times r)$.
If €$1 =$ £$r$, then €$X =$ £$(X \times r)$.
(i) €$1 =$ £$0.87$. Find the value of €$17$ in pounds.
Going from euro (base) to pounds — multiply.
$17 \times 0.87 = 14.79$
€$17 =$ £$14.79$
Try this: €$1 =$ £$0.87$. Find the value of €$50$ in pounds.
$50 \times 0.87 = 43.50$
£$43.50$
Try this: €$1 = \$ 1.06$. Find the value of €$250$ in dollars.
$250 \times 1.06 = 265$
$\$265$
Try this: €$1 =$ ¥$163.5$. Find the value of €$85$ in yen.
$85 \times 163.5 = 13{,}897.50$
¥$13{,}897.50$
Section 3
Converting to the base currency (divide)
When you're going to the base currency from the other, divide by the rate.
To base → divide:
If €$1 =$ £$r$, then £$Y =$ €$\dfrac{Y}{r}$.
If €$1 =$ £$r$, then £$Y =$ €$\dfrac{Y}{r}$.
(i) €$1 =$ £$0.87$. Find the value of £$29$ in euro.
Going from pounds to euro (base) — divide.
$\dfrac{29}{0.87} = 33.333\ldots$
£$29 =$ €$33.33$ (nearest cent)
Quick check: if the rate is less than $1$, dividing by it makes the answer bigger. £29 → €33.33 makes sense because euros are worth more than pounds in this example.
Try this: €$1 =$ £$0.87$. Find the value of £$100$ in euro.
$\tfrac{100}{0.87} = 114.94\ldots$
€$114.94$
Try this: €$1 = \$ 1.06$. Find the value of $\$ 500$ in euro.
$\tfrac{500}{1.06} = 471.69\ldots$
€$471.70$
Try this: €$1 =$ ¥$163.5$. Find the value of ¥$50{,}000$ in euro.
$\tfrac{50{,}000}{163.5} = 305.81\ldots$
€$305.81$
Section 4
Cross-rates: two currencies via a common base
Sometimes the two currencies you want to compare are both given in terms of a third currency (the base).
The trick is to set both sides equal to the base, then equate.
Bridge method:
1. Write each given rate as "(amount in currency X) $=$ $1$ unit of base".
2. Set the two sides equal (both equal the same $1$ unit of base).
3. Scale to find $1$ of one currency in terms of the other.
1. Write each given rate as "(amount in currency X) $=$ $1$ unit of base".
2. Set the two sides equal (both equal the same $1$ unit of base).
3. Scale to find $1$ of one currency in terms of the other.
(i) Given €$1 =$ £$0.95$ and €$1 = \$ 1.06$, find the rate of £ to \$ (i.e. £$1$ in dollars).
Rewrite both with euro on one side:
£$0.95 =$ €$1$
$\$ 1.06 =$ €$1$
Both equal €$1$, so:
£$0.95 = \$ 1.06$
Scale down to find £$1$: divide both sides by $0.95$.
£$1 = \dfrac{1.06}{0.95}$
$= 1.1157\ldots$
£$1 \approx \$ 1.12$
Sanity check: we know euros are worth less than dollars but more than pounds in this set-up. So £$1$ should be worth more than $\$ 1$. $\$ 1.12 > \$ 1$ ✓
Try this: €$1 =$ £$0.90$ and €$1 = \$ 1.10$. Find £$1$ in dollars.
£$0.90 = \$ 1.10$
£$1 = \tfrac{1.10}{0.90} = 1.222\ldots$
£$1 = \tfrac{1.10}{0.90} = 1.222\ldots$
£$1 \approx \$ 1.22$
Try this: €$1 = \$ 1.08$ and €$1 =$ ¥$160$. Find $\$ 1$ in yen.
$\$ 1.08 =$ ¥$160$
$\$ 1 = \tfrac{160}{1.08} = 148.15\ldots$
$\$ 1 = \tfrac{160}{1.08} = 148.15\ldots$
$\$ 1 \approx$ ¥$148.15$
Try this: €$1 =$ £$0.85$ and €$1 = \text{SFr}\, 0.96$ (Swiss francs). Find £$1$ in Swiss francs.
£$0.85 = \text{SFr}\, 0.96$
£$1 = \tfrac{0.96}{0.85} = 1.129\ldots$
£$1 = \tfrac{0.96}{0.85} = 1.129\ldots$
£$1 \approx \text{SFr}\, 1.13$
Try this (applied): a watch costs $\$ 240$ in New York. Given €$1 = \$ 1.06$, what does it cost in euro? Then, given also €$1 =$ £$0.87$, what would it cost a London shopper in pounds?
Price in €: $\tfrac{240}{1.06} = 226.42$
Price in £: $226.42 \times 0.87 = 196.99$
Price in £: $226.42 \times 0.87 = 196.99$
€$226.42$, then £$196.98$ (rounding)
End of lesson — Foreign Exchange.