Calculating a discount when getting better currency rates

Question

I have a hard time wrapping my head around this:

A particular item costs $10€$ in both shops.

Shop A and Shop B accept my currency $C$ as the payment method.

In Shop A, $1C$ is currently worth $10€.$ In Shop B, $1C$ is worth $13€,$ so essentially I'm getting $30$% more value with every purchase from Shop B (I think).

How would I calculate the discount that I are getting by spending my money in Shop B instead of Shop A?

What exchange rate should Shop B set (how much is $1C$ worth there) if its discount compared to Shop A is to be $-30\%\;?$

Answer 1

1. Shop B offers a $\dfrac{13-10}{13}=23\%$ discount over Shop A.

   (For example, a $130€$ items costs you $13C$ in Shop A but only $10C$ in Shop B.)

2. If Shop A's exchange rate remains unchanged, then Shop B offers a $30\%$ discount instead over Shop A only if its exchange rate is $1C:b\,€$ such that $$\frac{b-10}b=0.3\\b=100/7=14.29.$$

Answer 2

This is an application of dimensional analysis.

If the item is listed as $10€$ in both shops, then in terms of $C$ that works out to \begin{eqnarray} shopA:&&10€ = 10€ \frac{1C}{10€} = 1 C\\ shopB:&&10€ = 10€ \frac{1C}{13€} = \frac1{1.3} C \approx 0.77C \end{eqnarray} so every purchase of an item from shop A gives you, the seller, $1/(1/1.3) = 1.3$ times more $C$ than a purchase from shop B. Put another way, a purchase from shop A gives you 30% more than a purchase from shop B, which is the opposite of what you thought. If $1C = 10€$ at shop A but $1C = 13€$ at shop B, then $C$ is worth less (in €) at shop B, so for the same amount of € you get less $C$.

If you want to list the item as the same value $x$ in $C$ at both shops, then you convert for each shop separately\begin{eqnarray} shopA : &&xC = xC \cdot \frac{10\text{€}} {1C} = 10x€\\ shopB: &&xC = xC \cdot \frac{13€}{1C} = 13x€ \end{eqnarray} so you should list a price of 30% more € on shop B than on shop A.