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Jorge is evaluating two savings accounts for his \$2,000 investment. The first is a compound interest account with a 4\% annual rate compounded semi-annually, and the second is another compound interest account with the same annual rate but compounded quarterly. He prepared a table to show the total amount after 3 years but realized there is an error. Identify the mistake in Jorge's table.




Investment Type		Interest RateCompounding FrequencyTotal Amount after 3 Years

Semi-Annual Compounding		4\%Semi-Annually\$2,254.43

Quarterly Compounding		4\%Quarterly\$2,253.65






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


AThe quarterly compounding amount is incorrect.
BThe table is correct.
CThe semi-annual compounding amount is incorrect.

Verified Answer :

  • The compound interest formula is used to calculate the future value of the investments: \(A = P(1 + r/n)^{nt}\).
  • For semi-annual compounding, the calculated value is \(\$2,252.32\), while the table shows \(\$2,254.43\), indicating an error.
  • For quarterly compounding, the calculated value is \(\$2,253.65\), which matches the table.
  • The mistake is that the semi-annual compounding amount is incorrect. The final answer is the semi-annual compounding amount is incorrect.

Explanation

  1. Analyzing the Investment Accounts

To determine the correct totals for Jorge's investments, we use the compound interest formula:
\[A = P\left(1 + \frac{r}{n}\right)^{nt}\]
where \(P = \$2,000\) is the principal, \(r = 0.04\) is the annual interest rate, \(t = 3\) years, and \(n\) is the compounding frequency per year.

For the semi-annual account (\(n = 2\)):
\[A_{semi} = 2,000\left(1 + \frac{0.04}{2}\right)^{2 \times 3} = 2,000(1.02)^6\]
Calculating this:
\[2,000 \times 1.1261624 = 2,252.3248\]
Rounding to the nearest cent, the amount is \(\$2,252.32\). The table shows \(\$2,254.43\), which is incorrect.

For the quarterly account (\(n = 4\)):
\[A_{quart} = 2,000\left(1 + \frac{0.04}{4}\right)^{4 \times 3} = 2,000(1.01)^{12}\]
Calculating this:
\[2,000 \times 1.126825 \approx 2,253.65\]
The table shows \(\$2,253.65\), which is correct. Therefore, the error in the table lies in the semi-annual compounding amount.

Examples

Understanding compound interest is crucial for personal finance, such as comparing savings accounts or loans. By calculating the final balance with different compounding periods, individuals can maximize their returns on savings or minimize costs on debt. For example, if you are choosing between two high-yield savings accounts with the same interest rate, selecting the one with more frequent compounding will lead to a higher final balance over time. Mastering this formula helps you make more informed financial decisions, ensuring your money works as hard for you as possible.

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