Chapter 7, Problem 13CST

a.

To determine

The balance of each account after $\text{10}$ years.

Answer to Problem 13CST

For the account earning simple interest

Balance after $\text{10}$ years is $\text{\$750}$ .

For the account earning compound interest

Balance after $\text{10}$ years is $\$597.8$ .

Explanation of Solution

Given:

For the account earning simple interest

Principal ( $P$ ) = $\text{\$500}$

Annual interest rate ( $r$ ) = $5\%$

Time ( $t$ ) = $\text{10}$ years

For the account earning compound interest

Principal ( $P$ ) = $\text{\$350}$

Annual interest rate ( $r$ ) = $5.5\%$

Time ( $t$ ) = $\text{10}$ years

Concept Used:

Simple interest I is given by the formula

  $I=P\cdot r\cdot t$

Where $P$ is the principal, $r$ is the annual interest rate (written as a decimal), and $t$ is the time in years.

The balance of an account earning simple interest

  $A=P+Prt$

When an account earns interest compounded annually, the balance $A$ is given by the formula

  $A=P{(1+r)}^t$

Where $P$ is the principal, $r$ is the annual interest rate (written as a decimal), and $t$ is the time in years.

Compound interest is given by the formula

  $I=A-P$

Calculation:

As per the given problem

• For the account earning simple interest

Principal ( $P$ ) = $\text{\$500}$

Annual interest rate ( $r$ ) = $5\%$

Time ( $t$ ) = $\text{10}$ years

Write balance formula $A=P+Prt$

Substitute $500\text{ for }P,0.05\text{ for }r\text{ and 10 for }t$

  $\begin{array}{l}A=\text{ 500}+(500\cdot 0.05\cdot 10)\\ =500+250\text{ [multiply]}\\ \text{ = 750 [Add]}\end{array}$

Balance after $\text{10}$ years is $\text{\$750}$ .

• For the account earning compound interest

Principal ( $P$ ) = $\text{\$350}$

Annual interest rate ( $r$ ) = $5.5\%$

Time ( $t$ ) = $\text{10}$ years

Write balance formula $A=P{(1+r)}^t$

Substitute $350\text{ for }P,0.055\text{ for }r\text{ and 10 for }t$

  $\begin{array}{l}A=350{(}^1\\ =350{(}^{1.055}\text{ [use a calculator]}\\ =597.8\end{array}$

Balance after $\text{10}$ years is $\$597.8$ .

Conclusion:

For the account earning simple interest

Balance after $\text{10}$ years is $\text{\$750}$ .

For the account earning compound interest

Balance after $\text{10}$ years is $\$597.8$ .

b.

To determine

The total interest earned by each account after $\text{10}$ years.

Answer to Problem 13CST

The simple interest earned after $\text{10}$ years is $\$250$ .

The compound interest after $\text{10}$ years is $\$247.8$ .

Explanation of Solution

Given:

For the account earning simple interest

Principal ( $P$ ) = $\text{\$500}$

Annual interest rate ( $r$ ) = $5\%$

Time ( $t$ ) = $\text{10}$ years

For the account earning compound interest

Principal ( $P$ ) = $\text{\$350}$

Annual interest rate ( $r$ ) = $5.5\%$

Time ( $t$ ) = $\text{10}$ years

Concept Used:

Simple interest I is given by the formula

  $I=P\cdot r\cdot t$

Where $P$ is the principal, $r$ is the annual interest rate (written as a decimal), and $t$ is the time in years.

When an account earns interest compounded annually, the balance $A$ is given by the formula

  $A=P{(1+r)}^t$

Where $P$ is the principal, $r$ is the annual interest rate (written as a decimal), and $t$ is the time in years.

Compound interest is given by the formula

  $I=A-P$

Calculation:

As per the given problem

For the account earning simple interest

Write simple interest formula

  $I=P\cdot r\cdot t$

Substitute $500\text{ for }P,0.05\text{ for }r\text{ and 10 for }t$

I= $(500)(0.05)(10)$

  $=250\text{ [multiply]}$

Simple interest earned after $\text{10}$ years is $\$250$ .

For the account earning compound interest

Write compound interest formula

  $I=A-P$

Substitute $597.8$ for $A$ (calculated in the part (a)) and $350$ for $P$

  $\begin{array}{l}I=597.8-350\\ =247.8\text{ [Subtract]}\end{array}$

The compound interest after $\text{10}$ years is $\$247.8$ .

Conclusion:

The simple interest earned after $\text{10}$ years is $\$250$ .

The compound interest after $\text{10}$ years is $\$247.8$ .

c.

To determine

The compare the balance of the two accounts after $\text{10}$ years.

To compare the percent of change of balance of the two accounts.

Answer to Problem 13CST

The account earning simple interest has greater balance after $\text{10}$ years.

The percent of change of balance is greater for the account earning compound interest.

Explanation of Solution

Given:

For the account earning simple interest

Balance after $\text{10}$ years = $\text{\$750}$ .

For the account earning compound interest

Balance after $\text{10}$ years = $\$597.8$ .

(As calculated in part (a))

Concept used:

The percent of change is the ratio of the amount of increase or decrease to the original amount.

Percent of change,

  $p\%=\frac{\text{amount of increase or decrease}}{\text{original amount}}$

Calculation:

As per the given problem

For the account earning simple interest

Balance after $\text{10}$ years = $\text{\$750}$ .

For the account earning compound interest

Balance after $\text{10}$ years = $\$597.8$ .

As $\text{\$750>\$597}\text{.8}$ , so the account earning simple interest has greater balance after $\text{10}$ years.

Percent of change of balance for the account earning simple interest

  $p\%=\frac{\text{amount of increase or decrease}}{\text{original amount}}$

Amount of increase = Interest earned in $\text{10}$ years.

  $=\$250$

Substitute the values,

  $\begin{array}{l}p\%=\frac{\text{250 }}{500}\\ =\frac{1}{2}\text{ [cancel out common factors]}\\ \text{ =50\% [write as a percent]}\end{array}$

Therefore, percent of change in balance is $50\%$

Similarly,

Percent of change of balance for the account earning compound interest

Amount of increase = Interest earned in $\text{10}$ years.

  $=\$247.8$

Substitute the values,

  $\begin{array}{l}p\%=\frac{\text{247}\text{.8 }}{350}\\ =0.708\text{ [divide]}\\ \text{ =70}\text{.8\% [write as a percent]}\end{array}$

Therefore, percent of change in balance is $70.8\%$

Hence, percent of change of balance is greater for the account earning compound interest.

Conclusion:

The account earning simple interest has greater balance after $\text{10}$ years.

The percent of change of balance is greater for the account earning compound interest.