Question

(Annuity number of​ periods) ​You've just bought a new​ flat-screen TV for ​$3,400 and the store you bought it from offers to let you finance the entire purchase at an annual rate of 16 percent compounded monthly. If you take the financing and make monthly payments of ​$140.00​, how long will it take to pay off the​ loan? How much will you pay in interest over the life of the​ loan?

a. The number of years it will take to pay off the loan is
nothing years?

Answer 1

It will take approximately 32 months to pay off the loan, and the total interest paid over the life of the loan will be $1,080.

Step-by-step explanation:

To calculate the number of periods it will take to pay off the loan, we can use the formula for annuity:

N = log((PMT/i) * ((1 + i)^n)) / log(1 + i)

Where:

• N = number of periods
• PMT = monthly payment ($140)
• i = monthly interest rate (16%/12 = 0.0133)
• n = total number of payments

Plugging in the values, we get:

N = log((140/0.0133) * ((1 + 0.0133)^n)) / log(1 + 0.0133)

Using a scientific calculator or a spreadsheet software, we can solve for n. The result shows that it will take approximately 32 months to pay off the loan.

To calculate the total interest paid over the life of the loan, we can subtract the total amount borrowed ($3,400) from the total amount paid ($140 * 32 = $4,480).

Answer 2

9514 1404 393

Answer:

2.5 years

$736 in interest

Step-by-step explanation:

The relationship between the values in an amortization is ...

A = P(r/12)/(1 -(1 +r/12)^(-12t))

Here, we have A = 140, P = 3400, r = 0.16 and we want to find t

140 = 3400(0.16/12)/(1 -(1 +0.16/12)^(-12t))

1 -(1 +0.16/12)^(-12t) = 1/3.0882353

(1 +0.16/12)^(-12t) = 0.67619048

Taking logarithms gives ...

-12t·log(1 +0.16/12) = log(0.67619048)

t = log(0.67619048)/(-12·log(1+0.16/12)) ≈ 2.46177

It will take about 2.5 years to pay off the loan.

__

The total of payments is ...

(2.46177)(12)($140) = $4135.77

so, the interest paid is ...

$4135.77 -3400 = $735.77

You will pay about $736 in interest over the life of the loan.

_____

Additional comment

The number of payments is calculated to be about 29.54. You would probably make 30 payments, with the last payment being a different amount than $140. The actual amount will depend on when the payment is made, and the amount of it that is finance charge. We believe the remaining balance after 29 payments will be $75. After a full month, the finance charge due on that amount is $1.00, bringing the total amount of interest paid to $736.

If the loan is paid off in the middle of the month, the interest due is reduced. Above, we have calculated interest based on a last payment made part way through the month. We rounded up, partly because of the ambiguity in the final payment.