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Jack has $\$6,000$ to deposit for a term of 4 years. Bank C offers a $4.25\%$ annual interest rate compounded annually, while Bank D offers the same rate but compounded semi-annually.



What is the exact total amount Jack will have in each bank after 4 years?



Compound Interest Formula: $A=P\left(1+\frac{r}{n}\right)^{\text{nt}}$


ABank C: $\$7,150.00$, Bank D: $\$7,150.00$
BBank C: $\$7,200.00$, Bank D: $\$7,190.00$
CBank C: $\$7,086.89$, Bank D: $\$7,099.17$
DBank C: $\$7,158.22$, Bank D: $\$7,175.54$

Verified Answer :

  • Identify the principal \(P = \$6,000\), rate \(r = 0.0425\), and time \(t = 4\) years.
  • Apply the compound interest formula \(A = P(1 + r/n)^{nt}\) for both banks, using \(n=1\) for Bank C and \(n=2\) for Bank D.
  • Calculate the values to find \(A_C = \$7,086.89\) and \(A_D = \$7,099.17\).
  • Match the results with the given options, identifying the correct choice as \(\boxed{c}\).

Explanation

  1. Problem Analysis and Data identification

To determine the total amount Jack will have in each bank, we identify the given parameters: the principal amount \(P = \$6,000\), the time period \(t = 4\) years, and the annual interest rate \(r = 4.25\% = 0.0425\). We will use the compound interest formula \(A = P\left(1 + \frac{r}{n}\right)^{nt}\), where \(n\) is the number of compounding periods per year.

  1. Calculating Total Amount in Bank C

For Bank C, the interest is compounded annually, which means \(n = 1\). Plugging the values into the formula:
\[A_C = 6000\left(1 + \frac{0.0425}{1}\right)^{1 \times 4} = 6000(1.0425)^4\]
Calculating this, we find:
\[A_C = 6000 \times 1.181147825... \approx \$7,086.89\]

  1. Calculating Total Amount in Bank D

For Bank D, the interest is compounded semi-annually, which means \(n = 2\). Plugging the values into the formula:
\[A_D = 6000\left(1 + \frac{0.0425}{2}\right)^{2 \times 4} = 6000(1.02125)^8\]
Calculating this, we find:
\[A_D = 6000 \times 1.183195628... \approx \$7,099.17\]

  1. Conclusion

Comparing our results to the given options, we see that Bank C amounts to approximately \(\$7,086.89\) and Bank D amounts to approximately \(\$7,099.17\), which corresponds to option (c). Therefore, the exact total amounts after 4 years are \(\$7,086.89\) for Bank C and \(\$7,099.17\) for Bank D. The final answer is \(\boxed{c}\).

Examples

Understanding compound interest is a crucial skill for personal finance management. For example, if you are choosing a savings account or evaluating loan options, comparing the effects of different compounding frequencies—such as annually, semi-annually, or monthly—can significantly impact your long-term savings or debt. By using the compound interest formula \(A = P(1 + r/n)^{nt}\), you can make informed decisions about how your money grows over time, ensuring you select the best financial products for your needs.

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