Deborah borrows 5100 dollars from the bank at 5.6 percent interest rate compounded annually. The loan is to be repaid with 8 equal annual payments, the first coming a year from now. How large is each payment?

To determine how large each of Deborah's annual payments will be, we can use the **formula for the annuity payment** when a loan is being repaid with equal payments over time:

\[
A = \frac{P \cdot r}{1 - (1 + r)^{-n}}
\]

Where:
- $ A $ = the amount of each annual payment,
- $ P $ = the loan amount (principal),
- $ r $ = the interest rate per period,
- $ n $ = the number of periods (number of years).

### Given:
- $ P = 5100 $ dollars,
- $ r = 5.6\% = \frac{5.6}{100} = 0.056 $,
- $ n = 8 $ years.

Now, we can plug in the values and calculate $ A $.

\[
A = \frac{5100 \cdot 0.056}{1 - (1 + 0.056)^{-8}}
\]

Let's calculate that now.The size of each annual payment is approximately **$808.33**.

Would you like more details or have any questions about this calculation?

Here are 5 related questions to consider:
1. What happens to the payment size if the interest rate increases to 6%?
2. How would the payment change if the loan duration was 10 years instead of 8?
3. What is the total amount Deborah will have repaid after 8 years?
4. How much of the first payment goes towards interest versus the principal?
5. What is the outstanding balance after the first payment?

**Tip:** The formula for annuity payments can also be applied to saving plans where regular deposits are made at a constant interest rate.

Calculate the annual payment required to repay a $5100 loan at a 5.6% interest rate compounded annually over 8 years. This problem covers annuity payment calculations using the formula for equal loan repayments, suitable for students in Grades 11-12.

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