Question

In Exercises 19 and \(20,\) find the amount of time required for a \(\$ 2000\) investment to double if the annual interest rate \(r\) is compounded (a) annually, (b) monthly, (c) quarterly, and (d) continuously. \(r=8.25 \%\)

Short answer

The time it takes for the investment to double depends on the frequency of compounding. For annual compounding, it’ll take approximately 9 years, for monthly compounding it will take around 8.6 years, for quarterly compounding it will take about 8.7 years and for continuously compounding, the investment will double in approximately 8.4 years.

Step-by-step solution

Calculate for annual compounding

The formula for annual compound interest is \(A=P(1+r)^n\) where: \(A\) is the amount of money accumulated after n years, including interest.\(P\) is the principal amount (the initial amount of money)\(r\) is the annual interest rate (in decimal)\(n\) is the number of years the money is invested forSince the money is to double, \(A=2P\). Substituting this into the formula, we get \(2= (1 + 0.0825)^n\). Solve for \(n\)

Calculate for monthly compounding

The formula for compounded interest is \(A = P(1+ r/n) ^{nt}\), with n now representing the number of times the interest is compounded per year. Since the money is to be doubled, we substitute \(A=2P\) into the formula to get \(2= (1 + 0.0825/12)^{12n}\). Solve for \(n\)

Calculate for quarterly compounding

We still use the formula \(A = P(1+ r/n) ^{nt}\) but with \(n=4\) because the interest is compounded quarterly. We substitute \(A=2P\) into the formula to get \(2= (1 + 0.0825/4)^{4n}\). Solve for \(n\)

Calculate for continuous compounding

For continuously compounded interest, the formula is \(A = Pe^{rt}\), where \(e\) is Euler's number (\(2.71828\)). Substituting \(A=2P\) into the formula, we get \(2=e^{0.0825n}\). Solve for \(n\)