Chapter 3.4, Problem 142E

To determine

Is the time required for a continuously compoundedinvestment to quadruple twice as long as the time required for it to double? Give a reason for your answer and verify your answer algebraically.

Answer to Problem 142E

The time required for a continuously compounded investment to quadruple is twice as long as the time required for it to double.

Explanation of Solution

Concept used:

Continuous compound interest formula- $A=P{e}^{rt}$ , where,

A = Final amount after t years,

P = Principal amount,

r = Annual interest rate in decimal form.

Calculation:

Let us assume that the principal amount is P and annual interest rate is r.

The double of principal would be 2P.

Upon substituting our given values in continuous compound interest formula, we will get:

  $\begin{array}{l}A=P{e}^{rt}\left(\text{Continuous compound interest formula}\right)\\ 2P=P{e}^{rt}\left(\text{Substituting }A=2P\right)\\ \frac{2P}{P}=\frac{P{e}^{rt}}{P}\left(\text{Dividing both sides by }P\right)\\ 2=e^{rt}\\ e^{rt}=2\left(\text{Switching sides}\right)\\ \ln \left(e^{rt}\right)=\ln (2)\left(\text{Taking natural log on both sides}\right)\\ rt\ln (e)=\ln (2)\left(\text{Using property ln}\left(x^m\right)=m\ln (x)\right)\\ rt=\ln (2)\left(\text{Using property ln(}e)=1\right)\\ t=\frac{\ln (2)}{r}\end{array}$

The quadruple of principal would be 4P.

  $\begin{array}{l}A=P{e}^{rt}\left(\text{Continuous compound interest formula}\right)\\ 4P=P{e}^{rt}\left(\text{Substituting }A=4P\right)\\ \frac{4P}{P}=\frac{P{e}^{rt}}{P}\left(\text{Dividing both sides by }P\right)\\ 4=e^{rt}\\ e^{rt}=2^2\left(\text{Switching sides}\right)\\ \ln \left(e^{rt}\right)=\ln \left(2^2\right)\left(\text{Taking natural log on both sides}\right)\\ rt\ln (e)=2\ln (2)\left(\text{Using property ln}\left(x^m\right)=m\ln (x)\right)\\ rt=2\ln (2)\left(\text{Using property ln(}e)=1\right)\\ t=\frac{2\ln (2)}{r}\end{array}$

Upon comparing both times, we will get:

  $\frac{\text{Quadrupling time}}{\text{Doubling time}}=\frac{\frac{2\ln (2)}{r}}{\frac{\ln (2)}{r}}=\frac{2\ln (2)}{\ln (2)}=\frac{2}{1}$

Therefore, the time required for a continuously compounded investment to quadruple is twice as long as the time required for it to double.