Foreign Exchange

Section 1 of 4

Exchange rates — what they mean

An exchange rate tells you how much of one currency you get for one unit of another.

For example: $\text{€}1 = \text{£}0.87$.
This means "one euro is worth $0.87$ pounds".

Base currency: the one with $1$ beside it — the one we're quoting in.
In $\text{€}1 = \text{£}0.87$, 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 of 4

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 $\text{€}1 = \text{£}r$, then $\text{€}X = \text{£}(X \times r)$.

(i) $\text{€}1 = \text{£}0.87$. Find the value of $\text{€}17$ in pounds.

Going from euro (base) to pounds — multiply.

$17 \times 0.87 = 14.79$

$\text{€}17 = \text{£}14.79$

Try this: $\text{€}1 = \text{£}0.87$. Find the value of $\text{€}50$ in pounds.

$50 \times 0.87 = 43.50$

$\text{£}43.50$

Try this: $\text{€}1 = \$ 1.06$. Find the value of $\text{€}250$ in dollars.

$250 \times 1.06 = 265$

$\$265$

Try this: $\text{€}1 = \text{¥}163.5$. Find the value of $\text{€}85$ in yen.

$85 \times 163.5 = 13{,}897.50$

$\text{¥}13{,}897.50$

Section 3 of 4

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 $\text{€}1 = \text{£}r$, then $\text{£}Y = \text{€}\dfrac{Y}{r}$.

(i) $\text{€}1 = \text{£}0.87$. Find the value of $\text{£}29$ in euro.

Going from pounds to euro (base) — divide.

$\dfrac{29}{0.87} = 33.333\ldots$

$\text{£}29 = \text{€}33.33$ (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: $\text{€}1 = \text{£}0.87$. Find the value of $\text{£}100$ in euro.

$\dfrac{100}{0.87} = 114.94\ldots$

$\text{€}114.94$

Try this: $\text{€}1 = \$ 1.06$. Find the value of $\$ 500$ in euro.

$\dfrac{500}{1.06} = 471.69\ldots$

$\text{€}471.70$

Try this: $\text{€}1 = \text{¥}163.5$. Find the value of $\text{¥}50{,}000$ in euro.

$\dfrac{50{,}000}{163.5} = 305.81\ldots$

$\text{€}305.81$

Section 4 of 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.

(i) Given $\text{€}1 = \text{£}0.95$ and $\text{€}1 = \$ 1.06$, find the rate of £ to \$ (i.e. $\text{£}1$ in dollars).

Rewrite both with euro on one side:

$\text{£}0.95 = \text{€}1$

$\$ 1.06 = \text{€}1$

Both equal $\text{€}1$, so:

$\text{£}0.95 = \$ 1.06$

Scale down to find $\text{£}1$: divide both sides by $0.95$.

$\text{£}1 = \dfrac{1.06}{0.95}$

$= 1.1157\ldots$

$\text{£}1 \approx \$ 1.12$

Sanity check: we know euros are worth less than dollars but more than pounds in this set-up. So $\text{£}1$ should be worth more than $\$ 1$. $\$ 1.12 > \$ 1$ ✓

Try this: $\text{€}1 = \text{£}0.90$ and $\text{€}1 = \$ 1.10$. Find $\text{£}1$ in dollars.

$\text{£}0.90 = \$ 1.10$
$\text{£}1 = \dfrac{1.10}{0.90} = 1.222\ldots$

$\text{£}1 \approx \$ 1.22$

Try this: $\text{€}1 = \$ 1.08$ and $\text{€}1 = \text{¥}160$. Find $\$ 1$ in yen.

$\$ 1.08 = \text{¥}160$
$\$ 1 = \dfrac{160}{1.08} = 148.15\ldots$

$\$ 1 \approx \text{¥}148.15$

Try this: $\text{€}1 = \text{£}0.85$ and $\text{€}1 = \text{SFr}\, 0.96$ (Swiss francs). Find $\text{£}1$ in Swiss francs.

$\text{£}0.85 = \text{SFr}\, 0.96$
$\text{£}1 = \dfrac{0.96}{0.85} = 1.129\ldots$

$\text{£}1 \approx \text{SFr}\, 1.13$

Try this (applied): a watch costs $\$ 240$ in New York. Given $\text{€}1 = \$ 1.06$, what does it cost in euro? Then, given also $\text{€}1 = \text{£}0.87$, what would it cost a London shopper in pounds?

Price in $\text{€}$: $\dfrac{240}{1.06} = 226.42$
Price in $\text{£}$: $226.42 \times 0.87 = 196.99$

$\text{€}226.42$, then $\text{£}196.98$ (rounding)

End of lesson

Foreign Exchange — HL · Mathslive.ie

Combinations