Chapter 1.3, Problem 26E

To determine

To find: The time taken for an investment to double in value at the rate of $6.25\%$ compounded monthly.

Answer to Problem 26E

The time taken for an investment to double in value at the rate of $6.25\%$ compounded monthly is $11.1192$ years.

Explanation of Solution

Given information: A certain amount is invested at the rate of $6.25\%$ compounded monthly.

Calculation:

Assume that initial investment is $\$P$ and the interest rate is $6.25\%$ compounded monthly.

The formula for the value is:

  $y=P{\left(1+\frac{r}{100}\right)}^n$

The value of the amount is double the investment that is $2P$ . The interest is compounded monthly. So, the rate will become $\frac{1}{12}$ and time will become $12$ times.

Substitute $\frac{6.25}{12}$ for $r$ , $12t$ for $n$ and $\$2P$ for $y$ in the formula.

  $\begin{array}{c}y=P{\left(1+\frac{r}{100}\right)}^n\\ 2P=P{\left(1+\frac{6.25}{12\times 100}\right)}^{12t}\\ 2={\left(1+\frac{0.0625}{12}\right)}^{12t}\end{array}$

Follow the steps to solve the equation by calculator:

• First press “ON” button on graphical calculator and press $\boxed{\text{Y}=}$ key.
• Enter right hand side of the equation ${\left(1+\frac{0.0625}{12}\right)}^{12t}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{1}\boxed{+}\boxed{(}\boxed{0.0625}\boxed{÷}\boxed{12}\boxed{)}\boxed{)}\boxed{^}\boxed{(}\boxed{12}\boxed{\text{X}}\boxed{)}$ . Now, press the key $\boxed{\text{ENTER}}$ .
• Enter left hand side of the equation $2$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{2}$ .
• Press $\boxed{\text{WINDOW}}$ key and adjust the window at $\left[-10,20\right]$ by $\left[-10,20\right]$ .
• Now, press keystrokes given below to find the value of $t$ as:

  $\boxed{2\text{nd}}\to \boxed{\text{TRACE}}\to \boxed{5}\to \boxed{\text{ENTER}}\to \boxed{\text{ENTER}}\to \boxed{\text{ENTER}}$

  

So, the value of $\text{X}$ is $11.1192$ years.

Therefore, the time taken for an investment to double in value at the rate of $6.25\%$ compounded monthly is $11.1192$ years.