I was completely confused by that same explanation for a while. Don't even think of it like that. Think of it like this.
Definition: The effective rate of interest during the $n$th time period is
$$i_n = \frac{A(n) - A(n-1)}{A(n-1)}$$
Definition: The effective rate of discount during the $n$th time period is
$$d_n = \frac{A(n) - A(n-1)}{A(n)}$$
where $A(n)$ is the amount function (as defined in Kellison), the amount of money you have at time $n$. So, all you really need to understand here is that the rate of interest is a rate based on what you start with during the period. The rate of discount is a rate based on what you end up with. It's just two ways of looking at the same situation. There aren't a whole lot of real world situations where you borrow a bunch of money and immediately give some of it back. You would just borrow less.
By the way, that formula is all you need to calculate the effective rate of discount during period $n$ no matter what your $A(n)$ function is. So, in particular, it would work for your specific question of simple discount.
Question: Given a rate of 10% simple discount, calculate the effective rate of discount during period 5.
Answer: If we have 10% simple discount, then we know our accumulation function is $a(t) = \frac{1}{1 - 0.1t}$ for $0 \leq t < \frac{1}{d} = 10$. This is basically the definition of simple discount. If you have simple discount, this is your accumulation function. Memorize that. Then use it.
Therefore
$$d_5 = \frac{a(5) - a(4)}{a(5)} = \frac{2-10/6}{2} = \frac{1}{6} = 16.666666... \%$$
If you wanted to calculate the effective rate of interest when you are given the effective rate of simple discount, you can do that too. For example, in this same example,
$$i_5 = \frac{a(5) - a(4)}{a(4)} = \frac{2-10/6}{10/6} = \frac{1}{5} = 20 \%$$
Nothing changed. We're just looking at the same problem differently. In the discount case, how much money did we earn that period relative to how much we had at the end? In the interest case, how much money did we earn that period relative to how much we had at the beginning.
The formula you got is wrong. The correct present value formula given a future value $FV$ and with compounding $m$ times per period is:
$PV_{t=0}=\cfrac{FV_t}{(1+\cfrac{r}{m})^{t\cdot m}}$
I assume $r=8\%$ is the interest rate with annual compounding, and $t$ is measured in years, so $m=1$ and the formula simplifies to:
$PV_{t=0}=\cfrac{FV_t}{(1+r)^t}=FV_t(1+r)^{-t}$
$t=\cfrac{304}{365}$
(based on actual/actual convention)
So that:
$PV_{t=0}=10043.55(1+0.08\%)^{-\frac{304}{365}}=\$9,419.97$
Alternatively, if one understands the question like this: the bank wants to make a return of 8% in the given period, i.e. 8% is not an annualized rate, but the rate the bank applies for that specific period, then you would just have:
$P=FV(1+8\%)^{-1}$
which is also not the formula you have been using.
Best Answer
First of all you havn´t a perfect capital market, because you have transaction costs. In this case they are 85. The man has to pay the transaction costs. The man wants to borrow 21,915. He has to pay a fee of 85. Therefore he has to borrow in total 22,000. Let denote it $C_0$. After one year he has to pay back 25,000($C_1$).
Relation between $C_1, C_0$ and discount rate:
$C_0=(1-d)\cdot C_1$
Solving for d
$d=\frac{C_1-C_0}{C_1}=\frac{C_1-C_0}{C_1}\cdot 100\%$
In words:" How much is $C_0$ less than $C_1$ in relation to $C_1$.
$d=\frac{25,000-22,000}{25,000}\cdot 100\%=12\% $
Relation between $C_1, C_0$ and interest rate:
$C_1=(1+i)\cdot C_0$
Solving for i:
$i=\frac{C_1-C_0}{C_0}=\frac{C_1-C_0}{C_0}\cdot 100\% $
In words:" How much is $C_1$ bigger than $C_0$ in relation to $C_0$.
$i=\frac{25,000-22,000}{22,000}\cdot 100\%\approx 13.46\% $
The relation between i and d is:
$\boxed{d=\frac{i}{1+i}}$
Or
$i=\frac{d}{1-d}$