(Related **to Checkpoint**​ 6.1)  ​(Loan **amortization)**  On December​ 31, Beth Klemkosky bought a yacht for 

​$60 comma 00060,000. 

 She paid 

​$12 comma 00012,000 

down and agreed to pay the balance in 

1313 

equal annual installments that include both the principal and 

88 

percent interest on the declining balance.  How big will the annual payments​ be?

Question content area bottom

Part 1

**a.**  On December​ 31, Beth Klemkosky bought a yacht for 

​$60 comma 00060,000 

and paid 

​$12 comma 00012,000 

​down, how much does she need to borrow to purchase the​ yacht? 

​$48,00048,000  

​(Round to the nearest​ dollar.)

Part 2

**b.**  If Beth agrees to pay the loan plus 

88 

percent compound interest on the unpaid balance over the next 

1313 

years in 

1313 

equal​ end-of-year payments, what will those equal payments​ be? 

​$enter your response here  

​(Round to the nearest​ cent.)

### Part 1:
**a.** Beth Klemkosky purchased a yacht for $60,000 and made a down payment of $12,000. The amount she needs to borrow is:

\[
\text{Amount to borrow} = \text{Total price of yacht} - \text{Down payment}
\]

\[
\text{Amount to borrow} = 60,000 - 12,000 = 48,000
\]

So, the amount Beth needs to borrow is **$48,000**.

---

### Part 2:
**b.** To find Beth's equal annual payments, we will use the formula for **loan amortization with equal payments**, where the payments cover both the principal and interest. The formula for the payment is:

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

Where:
- $P$ = the annual payment
- $r$ = annual interest rate (in decimal form, so 8% becomes 0.08)
- $PV$ = present value of the loan (the amount borrowed, $48,000)
- $n$ = number of periods (in this case, 13 years)

#### Step-by-step calculation:

1. **Interest rate**: $r = 8\% = 0.08$
2. **Loan amount** (Present value, $PV$) = $48,000
3. **Number of payments (years)**, $n = 13$

Substitute these values into the amortization formula:

\[
P = \frac{0.08 \cdot 48,000}{1 - (1 + 0.08)^{-13}}
\]

First, calculate the denominator:

\[
1 + 0.08 = 1.08
\]

\[
1.08^{-13} \approx 0.3495
\]

Now, subtract that value from 1:

\[
1 - 0.3495 = 0.6505
\]

Now, calculate the payment $P$:

\[
P = \frac{0.08 \cdot 48,000}{0.6505}
\]

\[
P = \frac{3,840}{0.6505} \approx 5,902.81
\]

So, the annual payment Beth needs to make is approximately **$5,902.81**.

---

### Final Answers:
**a.** Amount to borrow: **$48,000**

**b.** Annual payment: **$5,902.81**

---

Would you like further details or have any other questions?

---

Here are 5 related questions to expand on this:
1. How is the interest portion of each payment calculated over time in an amortization schedule?
2. What is the total interest paid over the life of the loan?
3. How much of the principal is repaid by the end of the 5th year?
4. How would the payments change if the interest rate were 6% instead of 8%?
5. How does the loan amortization process differ for monthly payments instead of annual payments?

**Tip**: The earlier payments in an amortized loan are more heavily weighted towards paying off interest, while later payments focus more on reducing the principal.

On December 31, Beth Klemkosky bought a yacht for $60,000. She paid $12,000 down and agreed to pay the balance in 13 equal annual installments that include both the principal and 8% interest on the declining balance. How big will the annual payments be?

Learn how to calculate annual payments for a $48,000 loan with 8% interest over 13 years using the loan amortization formula. The problem covers key concepts such as principal, compound interest, and algebra. Suitable for high school and college students.

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