Chelsea invests $2,760 in a savings account with a fixed annual interest rate of 8% compounded 12 times per year. What will the account balance be after 6 years?
The correct answer and explanation is:
To find the account balance after 6 years with compound interest, we use the compound interest formula: A=P(1+rn)ntA = P \left(1 + \frac{r}{n} \right)^{nt}
Where:
- AA = future value of the investment (what we want to find)
- PP = principal amount (initial investment) = $2,760
- rr = annual interest rate = 8% = 0.08
- nn = number of times interest is compounded per year = 12
- tt = number of years = 6
Step-by-step Calculation:
A=2760(1+0.0812)12×6A = 2760 \left(1 + \frac{0.08}{12} \right)^{12 \times 6} A=2760(1+0.0066667)72A = 2760 \left(1 + 0.0066667 \right)^{72} A=2760(1.0066667)72A = 2760 \left(1.0066667 \right)^{72} A=2760×(1.558)A = 2760 \times (1.558) A \approx 2760 \times 1.558 \approx \boxed{$4,298.08}
Explanation (300 words):
Compound interest allows your money to grow faster than simple interest because you earn interest not only on the original amount you invested but also on the interest previously added. This is known as “interest on interest.”
In this example, Chelsea deposits $2,760 into a savings account with an annual interest rate of 8%, compounded monthly (12 times per year). Compounding monthly means that every month, interest is calculated and added to the balance, and future interest is calculated on the new balance.
Using the compound interest formula, we substitute the values:
- The interest rate per compounding period is 8%12=0.0066667\frac{8\%}{12} = 0.0066667
- The total number of compounding periods over 6 years is 12×6=7212 \times 6 = 72
By computing (1.0066667)72(1.0066667)^{72}, we get about 1.558, meaning the money grows by about 55.8% over 6 years. When we multiply this factor by the original $2,760, we find the final balance to be approximately $4,298.08.
This example shows how compound interest significantly increases savings over time, especially when the interest is compounded frequently and left to grow for multiple years. The more often it compounds, the more Chelsea earns.