Question

The municipality of Smallville has arranged to borrow​ $30 million in order to implement several public projects​ (flood control, school​ security, etc.). The interest rate will be​ 3% per​ year, payable at the end of each year. This​ $30 million debt will be retired by making payments of​ $5 million at the end of each year. The Board of Supervisors is concerned that it will take too long to pay off this debt. How many years will it take to retire this​ $30 million debt and its associated interest​ payments?

Answer 1

Answer:

The correct response is "6.71 years".

Explanation:

The given values are:

Loan amount

= $30 Million

Loan payment per year

= $5 Million

Interest rate

R = 3%

Let,

The take number of years will be "n".

⇒  $30=5\times (\frac{\frac{1-1}{(1+3 \ percent)^n}}{3 \ percent} )$

⇒  $30\times \frac{3 \ percent}{5}=(\frac{1-1}{1.03^n} )$

⇒  $.18=\frac{1-1}{1.03^n}$

⇒  $\frac{1}{1.03^n} =1-.18$

⇒  $1.03^n=\frac{1}{.82}=1.2195$

On taking log both sides, we get

⇒  $n=\frac{log (1.2195)}{log(1.03)}$

⇒  $n=6.71 \ years$