First, if the other fees are paid as part of the loan, I find a payment of 138.1625. If they are paid in advance, I believe they still count as interest. Then the effective interest is 159.95. The average balance is about half of the amount borrowed, as you start off owing 1498.5 and end at 0. So the effective annual interest rate is about 159.95/(1498.5/2)=23.15%. I think they get a lower value because they do an amortization, which keeps the balance higher for longer, but it is not far off. For the shorter terms, the interest rate goes higher because the fixed 39.95 fee gets spread over fewer months.
First, I will answer your second question. To reduce a number by 10%, you always multiply by 0.9. I don't know the context in which you think you were told that you sometimes divide by 1.1, but I am sure there is something here you have misunderstood. (I have a guess; since you asked about it, I have added it below.)
1: To remove 10% from 200, and then remove a compound 20% from that value:
- Remove 10% from 200, giving 200 - 200×10% = 200 - 20 = 180.
- Remove 20% from 180, giving 180 - 180×20% = 180 - 36 = 144.
The answer is 144.
2: Your idea here is correct.
3: To add 10% to 200, and then remove a compound 20% from that value:
- Add 10% to 200, giving 200 + 200×10% = 200 + 20 = 220.
- Subtract 20% from 220, giving 220 - 220×20% = 220 - 44 = 176.
4: You can use your idea from #2 here.
To handle multiple increases or reductions, just do them one step at a time.
I hope this is some help.
Suppose you see a price $p$ and you know that this price was marked up by 10% sometime in the past. You want to know what the original price was before the markup. This is $p ÷ 1.1$.
Notice that this is not the same as if you reduce $p$ by 10%! That would be $p × 0.9$, which is different.
I guess that the reason you thought you might reduce a number by 10% by dividing by 1.1 is that you were confused about this case. It seems as though reducing a number by 10% should be the same as undoing an increase of 10%, but it isn't.
For example, if you see a price of \$110 and you know it was marked up by 10%, then the original price was \$110 ÷ 1.1 = \$100. But to reduce \$110 by 10% you calculate 110 × 0.9 = \$99. Undoing a markup of 10% is not the same as reducing by 10%.
Best Answer
If $r$ is the fee (here $r=.03$) and $C$ is the charge you want to recover at the end, the amount to submit is $C/(1-r)$. So the mark-up rate is $r/(1-r)$, e.g., $.03/.97\approx.0309278$, or approximately $3.1\%$.
If you don't want to switch back and forth between percents and decimals, and prefer to think of the variable here as $r=3$, the pertinent formula for the mark-up rate is $100r/(100-r)$.