To solve this problem, we need to calculate the difference in monthly payments and the total interest when paying off the balance over one year rather than four years.

Given:

• Balance $P = 5800$
• Annual interest rate $r = 16\% = 0.16$
• Number of years for repayment $t = 4$ for the first case and $t = 1$ for the second case
• Number of payments per year $n = 12$ (monthly payments)
• Monthly payment for 4 years $PMT_{4 \text{ years}} = 164.40$

We will use the formula for the monthly payment of a loan:

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

Step 1: Calculate Monthly Payment for 1 Year

Substituting the values into the formula for the 1-year scenario:

• $P = 5800$
• $r = 0.16$
• $n = 12$
• $t = 1$

The equation becomes:

$PMT_{1 \text{ year}} = \frac{5800 \left(\frac{0.16}{12}\right)}{1 - \left(1 + \frac{0.16}{12}\right)^{-12}}$

Let's calculate this.The monthly payment for paying off the balance over 1 year is approximately $526.24. Therefore, you would need to pay about $361.84 more each month compared to the 4-year plan.

Next, let's calculate the total interest for each case and find the difference.

Step 2: Calculate Total Interest for Both Plans

The total amount paid over the loan duration is calculated as:

$\text{Total Amount Paid} = PMT \times n \times t$

The total interest is then:

$\text{Interest} = \text{Total Amount Paid} - P$

We'll calculate the total interest for both the 1-year and 4-year plans to determine how much less interest is paid when the balance is paid off in 1 year.The total interest paid over 1 year is approximately $514.87. When compared to the 4-year plan, you would save about $1576.33 in interest by paying off the balance in 1 year instead of 4 years.

Summary:

• Monthly payment for 1 year: $526.24
• Additional monthly payment: $361.84 more each month
• Total interest for 1 year: $514.87
• Interest savings: $1576.33 less in total interest

Would you like further details or have any questions?

Related Questions:

1. How would the monthly payment change if the interest rate increased to 18%?
2. What if you wanted to pay off the balance in 2 years instead of 4 or 1?
3. How does the length of the repayment period generally affect the total interest?
4. Can you calculate the break-even point when deciding between shorter or longer loan terms?
5. What if the monthly payment includes an additional fixed fee?

Tip:

Paying off debt faster reduces the amount of interest paid, but requires higher monthly payments. Balancing between affordability and interest savings is key.