Chapter 7.8, Problem 1GP

To determine

To write: A function to represent the amount earned by the first plan after $t$ years and then to write an equivalent function that represents the first plan with $4$ compoundings per year and then to determine effective quarterly rate of interest of the first plan and then determine the better plan.

Answer to Problem 1GP

  $\begin{array}{}$ A(t)=P{(1.05)}^t$\end{array}$

  $A(t)=P{(1.01227223443)}^{4t}$

  $EffectiveQuarterlyInterestRate(firstplan)=1.227223\%$

Second plan is better.

Explanation of Solution

Given information: First plan offers $5\%$ interest rate compounded annually, second plan offers $1.3\%$ interest rate compounded quarterly.

Calculation:

Compound interest formula $A=P{(1+\frac{r}{n})}^{nt}$ ,

Function representing the amount earned by the first plan after $t$ years,

  $\begin{array}{}$ A(t)=P{(1.05)}^t$\end{array}$

Equivalent function that represents the first plan with $4$ compoundings per year,

  $\begin{array}{}$ A(t)=P{(\sqrt[4]{1.05})}^{4t}$$ A(t)=P{(1.01227223443)}^{4t}$\end{array}$

Comparing above stated function with standard formula $A=P{(1+\frac{r}{n})}^{nt}$ ,

  $\begin{array}{l}1+\frac{r}{4}=1.01227223443\\ \\ EffectiveQuarterlyInterestRate=\frac{r}{4}=0.01227223443\end{array}$

  $EffectiveQuarterlyInterestRate=1.227223\%$

Second plan is better as its effective quarterly compounded interest rate is $1.3\%$ as compared to $1.227\%$ in first plan.