Question

rick took out a 10-year loan for $60,000 at an apr of 6%, compounded monthly. what will his balance be after he has made exactly half of his monthly payments?

a. $34,455.59

b. $20,032.62

c. $25,544.41

d. $39,967.38

Explanation:

Step1: Calculate the monthly interest rate and total number of payments

The annual percentage rate (APR) is \(6\% = 0.06\). The monthly interest rate \(r=\frac{0.06}{12}=0.005\).
The loan is for \(n = 10\) years. The total number of monthly payments \(N=10\times12 = 120\). Half - way through the payments, \(k = 60\) payments have been made.

Step2: Use the loan - balance formula

The loan - balance formula is \(B = P\times(1 + r)^{N}-M\times\frac{(1 + r)^{N}-(1 + r)^{k}}{r}\), where \(P\) is the principal amount, \(M\) is the monthly payment.
First, find the monthly payment \(M\) using the formula \(M=\frac{P\times r\times(1 + r)^{N}}{(1 + r)^{N}-1}\)
Substitute \(P = 60000\), \(r=0.005\), \(N = 120\) into the monthly - payment formula:

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Using a calculator, \((1.005)^{120}\approx1.8194\)

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Step3: Calculate the balance \(B\)

Now use the balance formula \(B = P\times(1 + r)^{N}-M\times\frac{(1 + r)^{N}-(1 + r)^{k}}{r}\)
Substitute \(P = 60000\), \(r = 0.005\), \(N=120\), \(k = 60\), \(M\approx666.12\)

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We know \((1.005)^{60}\approx1.3489\), \((1.005)^{120}\approx1.8194\)

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Answer:

A. \( \$34,455.59\)