Question

-Question 6, 5.2.19Find the interest rate for a $9500 deposit accumulating to $11,346, compounded annually for 7 years.The interest rate is_____%.(Do not round until the final answer. Then round to two decimal places as needed.

Answer 1

GIVEN

A compound interest account with a principal of $9500 accumulating to $11346 in 7 years, compounded annually.

TO FIND

The interest rate.

SOLUTION

The compound interest formula is given to be:

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

where

A = final amount

P = initial principal balance

r = interest rate

n = number of times interest applied per time period

t = number of time periods elapsed.

From the question, the following parameters are seen:

$\begin{gathered} A=11346 \\ P=9500 \\ n=1\text{ \lparen annual compounding\rparen} \\ t=7 \end{gathered}$

Therefore:

$\begin{gathered} 11346=9500(1+\frac{r}{1})^{1\times7} \\ 11346=9500(1+r)^7 \end{gathered}$

Solve for r:

$\begin{gathered} (1+r)^7=\frac{11346}{9500} \\ 1+r=\sqrt[7]{\frac{11346}{9500}} \\ r=\sqrt[7]{\frac{11346}{9500}}-1 \\ r=0.02569 \end{gathered}$

Multiply by 100:

$\begin{gathered} r=2.569\% \\ r\approx2.57\% \end{gathered}$

ANSWER

The interest rate required to get a total amount of $11,346.00 from compound interest on a principal of $9,500.00 compounded once per year over 7 years is 2.57% per year.