Q:

# A loan of $46,000 is made at 7% interest, compounded annually. After how many years will the amount due reach$65,000 or more?

Accepted Solution

A:
After 5.11 years, amount due reach $65,000 or moreSolution:Given that a loan of$46,000 is made. Rate of interest charged is 7% compounded annually Need to determine number of years in which the amount due reach $65,000 or more. In our case Amount due A =$65000 Loan Amount that is principal P = $46000 Rate of interest r = 7% Formula for Amount compounded anually is as follows:$$\mathrm{A}=P\left(1+\frac{r}{100}\right)^{n}$$Substituting the values in above formula we get$$\begin{array}{l}{65000=46000\left(1+\frac{7}{100}\right)^{n}} \\\\ {\frac{65000}{46000}=\left(\frac{107}{100}\right)^{n}} \\\\ {\Rightarrow 1.4130=(1.07)^{n}}\end{array}$$Applying log on both sides, we$$\begin{array}{l}{\ln 1.4130=n \ln 1.07} \\\\ {=>\frac{\ln 1.4130}{\ln 1.07}=\mathrm{n}} \\\\ {=>\mathrm{n}=\frac{0.34571}{0.067658}=5.1096=5.11}\end{array}$$Hence after 5.11 years , amount due reach$65,000 or more