While looking at exchange rates for an upcoming vacation, I decided to brush up on some old math but wanted to make sure I was thinking about it correctly.
$B_1$ charges a rate for USD to EUR conversion of $0.85$ but charges a shipping fee ($F$). I want to calculate the effective rate of $B_1 (B_1')$ with a given $F$ for an amount $A$.
$B_1'*(A+F) = B_1*A$
$or
$B_1' = \dfrac{B_1*A}{A+F}$
So for $F = 10$ and $A = 500$
$B_1' = \dfrac{425}{510}$ or $B_1' = 0.833$
Is this the right way to think about it?
Secondly another bank, $B_2$ offers a lower initial rate of $0.83$ but has no fee. How would one construct a formula to determine the value of $A$ for a given $F$ at which $B_1'$ and $B_2$ are even (practically, when to go with one over the other)?
Well you have $$ B_2 = B_1' = \frac{B_1 \cdot A}{A+F} $$ which you need to solve for $A$: $$ B_2 (A+F) = B_1 A \\ B_2 F = B_1 A - B_2 A\\ A = \frac{B_2 F}{B_1 - B_2} $$