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Naily [24]
2 years ago
15

(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 le

t 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?
Mathematics
1 answer:
AleksAgata [21]2 years ago
4 0

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.

_____

<em>Additional comment</em>

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.

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