Chapter 7.7, Problem 31HP

To determine

To calculate:Time taken for Santos’s investment to reach $2500.

Answer to Problem 31HP

It will take 9.21 years for Santos’s investment to reach $2500.

Explanation of Solution

Given information:Santos invested $1200 into an account with an interest rate of 8% compounded monthly.

Formula used:

Compound interest formula for compounded monthly is given by,

  $A=P{(1+\frac{\raisebox{1ex}{$R$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}{100})}^{12t}$

where, $A=$ Amount

  $P=$ Principal

  $R=$ Rate % per annum

  $t=$ time

Calculation:

Here, $A=$ $2500, $P=$ $1200, $R=8\%$ .

Then, time taken for the amount to reach $2500,

  $A=P{(1+\frac{\raisebox{1ex}{$R$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}{100})}^{12t}$

  $\Rightarrow 2500=1200{(1+\frac{\raisebox{1ex}{$8$}\!\left/ \!\raisebox{-1ex}{$12$}\right.}{100})}^{12t}$

  $\Rightarrow 2500=1200{(1+\frac{8}{1200})}^{12t}$

  $\Rightarrow \frac{2500}{1200}={(\frac{1208}{1200})}^{12t}$

Taking log on both sides,

  $\begin{array}{l}$ \log (\frac{2500}{1200})=\log {(\frac{1208}{1200})}^{12t}$\Rightarrow 12t\log (\frac{1208}{1200})=\log (\frac{2500}{1200})\\ \Rightarrow 12t\times 0.0028856883=0.3187587626\\ \Rightarrow 12t=\frac{0.3187587626}{0.0028856883}\\ \Rightarrow 12t=110.461952041\\ \Rightarrow t=9.2051626701\end{array}$

Rounding to nearest hundredths, $t=9.21$

Thus, it will take 9.21 years for Santos’s investment to reach $2500.