Question

Find the interest rate for a $7000 deposit accumulating $7789, compounded annually for six years.The interest rate is ??%

Answer 1

The rule of the compounded interest is

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

Where:

A is the new value

P is the initial value

r is the rate in decimal

n is the number of periods per year

t is the time in years

Since the deposit amount is $7000

P = 7000

Since it is accumulating $7789

A = 7789

Since the rate is compounded annually for six years

n = 1

t = 6

Let us substitute these values in the rule above to find r

$\begin{gathered} 7789=7000(1+\frac{r}{1})^{1\times6} \\ 7789=7000(1+r)^6 \end{gathered}$

Divide both sides by 7000

$\begin{gathered} \frac{7789}{7000}=\frac{7000(1+r)^6}{7000} \\ \frac{7789}{7000}=(1+r)^6_{} \end{gathered}$

To solve this we will insert a log for each side or find root 6 for both sides

$\begin{gathered} \sqrt[6]{\frac{7789}{7000}}=\sqrt[6]{(1+r)^6} \\ 1.01795976=1+r \end{gathered}$

Subtract 1 from both sides to find r

$\begin{gathered} 1.01795976-1=1-1+r \\ 0.01795976=r \end{gathered}$

Now, we multiply it by 100% to change it to a percentage

r = 0.01795976 x 100%

r = 1.795976

We can round it to the nearest 2 decimal places

r = 1.80%