Question

Betty has $134 in a savings account. The interest rate is 2%, compounded annually.To the nearest cent, how much interest will she earn in 3 years?

Answer 1

You have to calculate the interest Betty will earn after three years, if she has an initial amount of $134 and the account compounds annually at a 2% interest rate.

To determine the total interest after three years, the first step is to calculate the accrued amount. We know that the accrued amount after a t period of times is equal to the principal amount (or initial amount) plus the interest

$A=P+I$

A= accrued amount

P= principal amount

I= interest

To calculate the accrued amount of an account with compound interest you have to use the following formula:

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

Where

A= accrued amount

P= principal amount

r= interest rate expressed as a decimal value

t= time period involved, expressed as years

n= number of compounding periods per unit of t

Compounding periods

This account compounds annually, which means that there is one compound period per year, If the time is 3 years, the total number of compounding periods will be:

$\begin{gathered} n=1\frac{period}{\text{year}}\cdot3\text{years} \\ n=3\text{periods} \end{gathered}$

To express the interest rate as a decimal value you have to divide the percentage by 100:

$r=\frac{2}{100}=0.02$

The principal amount is P=134, and the time period is t=3, so the accrued amount can be calculated as:

$\begin{gathered} A=134(1+\frac{0.02}{3})^{3\cdot3} \\ A=134(1+\frac{1}{150})^9 \\ A=142.26 \end{gathered}$

The accrued amount after 3 years will be $142.26

To determine the interest you have to subtract the principal amount from the accrued amount:

$\begin{gathered} I=A-P \\ I=142.26-134 \\ I=8.26 \end{gathered}$

So the total interest after 3 years will be I=$8.26