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+(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+(r)/(1))^(1*6) \\ 7789=7000(1+r)^6 \end{gathered}`

Divide both sides by 7000

`\begin{gathered} (7789)/(7000)=(7000(1+r)^6)/(7000) \\ (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]{(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%