To solve this problem, we will use the formula for compound interest:

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

Where:

• $A$ is the amount owed after $t$ years.
• $P$ is the principal amount (initial amount borrowed).
• $r$ is the annual interest rate (as a decimal).
• $n$ is the number of times the interest is compounded per year.
• $t$ is the number of years.

In this case:

• $P = 3500$ dollars
• $r = 0.13$ (since 13% is equivalent to 0.13 as a decimal)
• $n = 1$ (since the interest is compounded annually)
• $t = 1$ for part (A) and $t = 2$ for part (B).

Part (A) - Amount Owed at the End of Year 1

For $t = 1$:

$A_1 = 3500 \left(1 + \frac{0.13}{1}\right)^{1 \times 1}$

Simplifying:

$A_1 = 3500 \left(1 + 0.13\right)^{1}$

$A_1 = 3500 \times 1.13$

$A_1 = 3955$

So, the amount owed at the end of year 1 is $3955.

Part (B) - Amount Owed at the End of 2 Years

For $t = 2$:

$A_2 = 3500 \left(1 + \frac{0.13}{1}\right)^{1 \times 2}$

Simplifying:

$A_2 = 3500 \times 1.13^2$

$A_2 = 3500 \times 1.2769$

$A_2 = 4469.15$

So, the amount owed at the end of 2 years is $4469.15.

Summary:

• (A) The amount owed at the end of year 1 is $3955.
• (B) The amount owed at the end of year 2 is $4469.15.

Would you like more details or have any questions?

Here are some related questions:

1. How would the amount change if the interest were compounded quarterly?
2. What if the interest rate were 10% instead of 13%?
3. How would the total interest be calculated after 2 years?
4. What is the formula to find the amount owed after 3 years?
5. How does changing the compounding frequency affect the amount owed?
6. Can you calculate the amount owed after 5 years with the same conditions?
7. What is the effective annual rate (EAR) for this loan?
8. How would the amount differ if payments were made annually?

Tip: When dealing with compounded interest, always remember to adjust the interest rate and the time period according to the compounding frequency.