Error AnalysisYou Deposit $ 250 \$250 $250 In An Account That Pays 1.25 % 1.25\% 1.25% Annual Interest. Describe And Correct The Error In Finding The Balance After 3 Years When The Interest Is Compounded Quarterly.Given Calculation:[A =

Understanding the Problem

When dealing with compound interest, it's essential to understand the formula and the assumptions made. In this case, we have an initial deposit of $\$250$ and an annual interest rate of $1.25\%$. The interest is compounded quarterly, meaning that the interest is applied four times a year.

The Given Calculation

The given calculation is as follows:

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

Where:

• $A$ is the future value of the investment/loan, including interest
• $P$ is the principal investment amount (the initial deposit or loan amount)
• $r$ is the annual interest rate (in decimal)
• $n$ is the number of times that interest is compounded per year
• $t$ is the time the money is invested or borrowed for, in years

The Error in the Calculation

The error in the calculation is that the interest rate is not converted to a decimal. The interest rate is given as $1.25\%$, but it should be converted to a decimal by dividing by $100$. This would give us an interest rate of $0.0125$.

Correcting the Error

To correct the error, we need to convert the interest rate to a decimal and then plug in the values into the formula.

${A = 250 \left(1 + \frac{0.0125}{4}\right)^{4 \cdot 3}}$

${A = 250 \left(1 + 0.003125\right)^{12}}$

${A = 250 \left(1.003125\right)^{12}}$

${A = 250 \cdot 1.039}$

${A = 259.75}$

Conclusion

In conclusion, the error in the calculation was due to not converting the interest rate to a decimal. By correcting this error, we were able to find the correct balance after 3 years when the interest is compounded quarterly.

Understanding Compound Interest

Compound interest is a type of interest that is calculated on both the initial principal and the accumulated interest from previous periods. It's a powerful tool for growing your savings over time, but it can also be complex to understand.

Key Concepts in Compound Interest

• Principal: The initial amount of money deposited or borrowed.
• Interest Rate: The rate at which interest is applied to the principal.
• Compounding Frequency: The number of times that interest is applied per year.
• Time: The length of time that the money is invested or borrowed for.

Formulas for Compound Interest

The formula for compound interest is:

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

Where:

• $A$ is the future value of the investment/loan, including interest
• $P$ is the principal investment amount (the initial deposit or loan amount)
• $r$ is the annual interest rate (in decimal)
• $n$ is the number of times that interest is compounded per year
• $t$ is the time the money is invested or borrowed for, in years

Examples of Compound Interest

• Quarterly Compounding: If interest is compounded quarterly, the formula becomes: ${A = P \left(1 + \frac{r}{4}\right)^{4t}}$
• Monthly Compounding: If interest is compounded monthly, the formula becomes: ${A = P \left(1 + \frac{r}{12}\right)^{12t}}$

Conclusion

In conclusion, compound interest is a powerful tool for growing your savings over time. However, it can also be complex to understand. By understanding the key concepts and formulas, you can make informed decisions about your investments and loans.

Common Mistakes in Compound Interest

• Not converting the interest rate to a decimal: This is a common mistake that can lead to incorrect calculations.
• Not understanding the compounding frequency: This can lead to incorrect calculations and a misunderstanding of the interest rate.
• Not considering the time: This can lead to incorrect calculations and a misunderstanding of the interest rate.

Conclusion

Frequently Asked Questions

Q: What is compound interest?

A: Compound interest is a type of interest that is calculated on both the initial principal and the accumulated interest from previous periods.

Q: How does compound interest work?

A: Compound interest works by applying the interest rate to the principal amount, and then adding the interest to the principal amount. This process is repeated for each compounding period, resulting in a snowball effect that grows the principal amount over time.

Q: What are the key factors that affect compound interest?

A: The key factors that affect compound interest are:

• Principal: The initial amount of money deposited or borrowed.
• Interest Rate: The rate at which interest is applied to the principal.
• Compounding Frequency: The number of times that interest is applied per year.
• Time: The length of time that the money is invested or borrowed for.

Q: What is the formula for compound interest?

A: The formula for compound interest is:

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

Where:

• $A$ is the future value of the investment/loan, including interest
• $P$ is the principal investment amount (the initial deposit or loan amount)
• $r$ is the annual interest rate (in decimal)
• $n$ is the number of times that interest is compounded per year
• $t$ is the time the money is invested or borrowed for, in years

Q: What is the difference between simple interest and compound interest?

A: Simple interest is calculated only on the principal amount, whereas compound interest is calculated on both the principal and the accumulated interest from previous periods.

Q: How often is interest compounded?

A: Interest can be compounded daily, monthly, quarterly, or annually, depending on the type of investment or loan.

Q: What is the effect of compounding frequency on compound interest?

A: The compounding frequency has a significant impact on compound interest. The more frequently interest is compounded, the faster the principal amount grows.

Q: Can compound interest be negative?

A: Yes, compound interest can be negative. This occurs when the interest rate is less than the inflation rate, resulting in a decrease in the purchasing power of the principal amount.

Q: How can I calculate compound interest?

A: You can calculate compound interest using a compound interest calculator or by using the formula:

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

Q: What are some common mistakes to avoid when calculating compound interest?

A: Some common mistakes to avoid when calculating compound interest include:

• Not converting the interest rate to a decimal: This can lead to incorrect calculations.
• Not understanding the compounding frequency: This can lead to incorrect calculations and a misunderstanding of the interest rate.
• Not considering the time: This can lead to incorrect calculations and a misunderstanding of the interest rate.

Conclusion

In conclusion, compound interest is a powerful tool for growing your savings over time. By understanding the key concepts and formulas, you can make informed decisions about your investments and loans.