Chapter 6.6, Problem 41STP

To determine

To find: Which Plan will produce greater earnings and how much more than other plan.

Answer to Problem 41STP

Earnings through Plan B is $\$16.29$ greater than Earnings through Plan A.

Explanation of Solution

Given information:

Principal amount $=\$500$

Simple interest percent $=6.75\%$ .

Compound Interest percent $=6.5\%$

Time $=5$ years

It is compounded annually so, value of $n$ is $1$ .

Concept:

Simple interest:

  $I=prt$

Where

  $I=$ Simple interest.

  $p=$ Principal amount

  $t=$ time in years.

  $r=$ Annual interest rate.

Compound interest:

  $A=p{\left(1+\frac{r}{n}\right)}^{nt}$

Where

  $A=$ Total amount

  $p=$ Principal amount

  $t=$ time in years.

  $r=$ Annual interest rate.

  $n=1$ for compounded annually.

Calculation:

Using simple interest From given information

Time $=t=5$ years.

Principal amount $=p=\$500$ .

Interest percent $=R=6.75\%$

We know

Annual interest rate $=r=\frac{R}{100}$

  $\begin{array}{l}\Rightarrow r=\frac{6.75}{100}\\ \Rightarrow r=0.0675\end{array}$

On substituting values in above mentioned concept

We get

  $\begin{array}{l}I=prt\\ \Rightarrow I=\left(500\right)\left(0.0675\right)\left(5\right)\\ \Rightarrow I=168.75\end{array}$

Therefore,

Final amount $=$ Principal amount $+$ Simple interest.

  $\Rightarrow$ Final amount $=$$500+168.75=\$668.75$

Therefore,

Final amount using simple interest is $\$668.75$

Using Compound interest From given information

Time $=t=5$ years.

Principal amount $=p=\$500$ .

Interest percent $=r=6.5\%=0.065$

On substituting values in above mentioned concept

We get

  $\begin{array}{l}A=p{\left(1+\frac{r}{n}\right)}^{nt}\\ \Rightarrow A=500{\left(1+\frac{0.065}{1}\right)}^{5\left(1\right)}\\ \Rightarrow A=500{\left(1+0.065\right)}^5\\ \Rightarrow A=500{\left(1.065\right)}^5\\ \Rightarrow A=500\left(1.37008666\right)\\ \Rightarrow A=\$685.04333\end{array}$

Therefore,

To nearest cent, Total amount using compound interest $=A=\$685.04$

We got

Final amount using simple interest is $\$668.75$

Total amount using compound interest $=A=\$685.04$

Therefore,

On comparing both values

We can say that

PlanB using compound interest there will be greater earnings than Plan A.

Difference of earnings $=$ Plan B $-$ Plan A

  $\Rightarrow$ Difference of earnings $=$$685.04-668.75$

  $\Rightarrow$ Difference of earnings $=$$\$16.29$

Therefore,

Earnings through Plan B is $\$16.29$ greater than Earnings through Plan A.