Bob signs a note promising to pay Marie $3500 in 7 years at 11.5% compounded monthly. Then, 46 days before the note is due, Marie sells the note to a bank which discounts the note based on a bank discount rate of 17.5%. How much did the bank pay Marie for the note?

We are asked to calculate how much the bank paid Marie for a note that Bob signed, promising to pay $3,500 in 7 years, with an interest rate of 11.5% compounded monthly. Then Marie sells this note to the bank 46 days before the note's due date, with the bank applying a discount rate of 17.5%.

### Step 1: Future Value of the Note
First, we need to calculate the future value of the note, $ A $, which is the amount Bob will pay at the end of 7 years.

The formula for future value with compound interest is:

\[
A = P \left( 1 + \frac{r}{n} \right)^{nt}
\]

Where:
- $ P = 3500 $ is the principal.
- $ r = 11.5\% = 0.115 $ is the annual interest rate.
- $ n = 12 $ is the number of times the interest is compounded per year (monthly compounding).
- $ t = 7 $ years is the time period.

Now, let's calculate the future value.

\[
A = 3500 \left( 1 + \frac{0.115}{12} \right)^{12 \times 7}
\]

### Step 2: Discounting the Note
Marie sells the note 46 days before it is due. First, we need to find how much time is left before the note is due in terms of years. Since there are 365 days in a year:

\[
\text{Remaining time} = \frac{46}{365} \text{ years}
\]

The bank discounts the note based on a discount rate of 17.5%. The formula for the bank discount (also known as the present value of a future sum) is:

\[
PV = A \left( 1 - d \times t_{\text{remaining}} \right)
\]

Where:
- $ PV $ is the present value (the amount the bank pays Marie).
- $ A $ is the future value we calculated in step 1.
- $ d = 17.5\% = 0.175 $ is the bank's discount rate.
- $ t_{\text{remaining}} $ is the remaining time (46 days in years).

Now, let's proceed with the calculation.### Step 1: Future Value of the Note
The future value of the note, after 7 years at 11.5% compounded monthly, is approximately:

\[
A = 7798.49 \, \text{USD}
\]

### Step 2: Present Value (Amount Paid by the Bank)
The amount the bank paid Marie for the note, after applying a 17.5% discount with 46 days remaining, is approximately:

\[
PV = 7626.50 \, \text{USD}
\]

So, the bank paid Marie **$7,626.50** for the note.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How does compounding frequency affect the future value of an investment?
2. What is the difference between simple interest and compound interest?
3. How would the present value change if the bank discount rate was 20%?
4. How would the payment change if Marie sold the note 90 days before the due date?
5. What are other applications of the bank discount rate in finance?

**Tip:** The higher the frequency of compounding, the more interest will accumulate over time.

Learn how to calculate the amount a bank pays for a note using a 17.5% discount rate and compound interest at 11.5%. This step-by-step guide explains the formulas for future value and bank discount for a $3500 note, compounded monthly over 7 years. Ideal for high school students.

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