Question

Lush Gardens Co. bought a new truck for $50,000. It paid $6,000 of this amount as a down payment and financed the balance at 4.80% compounded semi-annually. If the company makes payments of $1,500 at the end of every month, how long will it take to settle the loan?

Answer 1

To calculate the time it will take to settle the loan, we need to consider the monthly payments and the interest rate. Let's break down the steps:

1. Loan amount: The loan amount is the purchase price minus the down payment:

Loan amount = $50,000 - $6,000 = $44,000

2. Calculate the monthly interest rate: The annual interest rate of 4.80% compounded semi-annually needs to be converted to a monthly rate. Since interest is compounded semi-annually, we have 2 compounding periods in a year.

Monthly interest rate = (1 + annual interest rate/2)^(1/6) - 1

Monthly interest rate = (1 + 0.0480/2)^(1/6) - 1 = 0.03937

3. Calculate the number of months needed to settle the loan using the monthly payment and interest rate. We can use the formula for the number of months needed to pay off a loan:

n = -log(1 - r * P / M) / log(1 + r),

where:

n = number of periods (months),

r = monthly interest rate,

P = loan amount,

M = monthly payment.

Plugging in the values:

n = -log(1 - 0.03937 * $44,000 / $1,500) / log(1 + 0.03937)

Calculating this expression, we find:

n ≈ 30.29

Therefore, it will take approximately 30.29 months to settle the loan.

Hope it helps!