Suppose $$ 6 , 500 6,500 6 , 500 $ Is Invested In An Account That Earns Interest At A Rate Of $2%$ Compounded Quarterly For 10 Years. Describe And Correct The Error A Student Made When Finding The Value Of The Account. [
Introduction
Compound interest is a fundamental concept in mathematics, particularly in finance and economics. It refers to the interest earned on both the principal amount and any accrued interest over time. In this article, we will explore a scenario where a student made an error in calculating the value of an account with compound interest. We will describe the error, explain the correct formula, and provide a step-by-step solution to the problem.
The Problem
Suppose $6,500 is invested in an account that earns interest at a rate of 2% compounded quarterly for 10 years. A student attempted to find the value of the account using the compound interest formula, but made an error. The formula for compound interest is:
A = P(1 + r/n)^(nt)
Where:
- A is the amount of money accumulated after n years, including interest
- P is the principal amount (initial investment)
- r is the annual interest rate (in decimal form)
- n is the number of times that interest is compounded per year
- t is the time the money is invested for in years
The Student's Error
The student incorrectly applied the formula, resulting in an incorrect value for the account. Let's examine the student's work and identify the error.
Student's Work
The student wrote:
A = 6500(1 + 0.02/4)^(4*10) A = 6500(1 + 0.005)^40 A = 6500(1.005)^40 A = 6500 * 1.2214 A = 7,934.10
Error Analysis
The student's error lies in the calculation of the interest rate per period (r/n). The interest rate is 2% per annum, but it is compounded quarterly. Therefore, the interest rate per period should be 0.02/4 = 0.005, not 0.005^2.
Correct Solution
To find the correct value of the account, we will apply the compound interest formula correctly.
A = P(1 + r/n)^(nt) A = 6500(1 + 0.02/4)^(4*10) A = 6500(1 + 0.005)^40 A = 6500(1.005)^40 A = 6500 * 1.2214 A = 7,943.10
Conclusion
In this article, we identified and corrected an error made by a student when calculating the value of an account with compound interest. The student incorrectly applied the formula, resulting in an incorrect value. We explained the correct formula and provided a step-by-step solution to the problem. By understanding the compound interest formula and applying it correctly, we can accurately calculate the value of an account with compound interest.
Compound Interest Formula
The compound interest formula is:
A = P(1 + r/n)^(nt)
Where:
- A is the amount of money accumulated after n years, including interest
- P is the principal amount (initial investment)
- r is the annual interest rate (in decimal form)
- n is the number of times that interest is compounded per year
- t is the time the money is invested for in years
Example Calculations
Let's consider a few examples to illustrate the compound interest formula.
Example 1
Suppose $5,000 is invested in an account that earns interest at a rate of 3% compounded annually for 5 years. What is the value of the account?
A = P(1 + r/n)^(nt) A = 5000(1 + 0.03/1)^(1*5) A = 5000(1 + 0.03)^5 A = 5000(1.03)^5 A = 5000 * 1.1593 A = 5,796.50
Example 2
Suppose $10,000 is invested in an account that earns interest at a rate of 4% compounded quarterly for 10 years. What is the value of the account?
A = P(1 + r/n)^(nt) A = 10000(1 + 0.04/4)^(4*10) A = 10000(1 + 0.01)^40 A = 10000(1.01)^40 A = 10000 * 2.2083 A = 22,083.00
Conclusion
Introduction
Compound interest is a fundamental concept in mathematics, particularly in finance and economics. In our previous article, we explored a scenario where a student made an error in calculating the value of an account with compound interest. We explained the correct formula and provided a step-by-step solution to the problem. In this article, we will answer some frequently asked questions about compound interest.
Q&A
Q1: What is compound interest?
A1: Compound interest is the interest earned on both the principal amount and any accrued interest over time. It is a type of interest that is calculated on the initial principal, which also includes all the accumulated interest from previous periods.
Q2: What is the formula for compound interest?
A2: The formula for compound interest is:
A = P(1 + r/n)^(nt)
Where:
- A is the amount of money accumulated after n years, including interest
- P is the principal amount (initial investment)
- r is the annual interest rate (in decimal form)
- n is the number of times that interest is compounded per year
- t is the time the money is invested for in years
Q3: What is the difference between simple interest and compound interest?
A3: Simple interest is calculated only on the principal amount, whereas compound interest is calculated on both the principal amount and any accrued interest. Compound interest is more beneficial to the investor as it earns interest on interest.
Q4: How often is interest compounded?
A4: Interest can be compounded daily, monthly, quarterly, or annually, depending on the investment or loan agreement.
Q5: What is the effect of compounding frequency on interest earned?
A5: The more frequently interest is compounded, the more interest is earned. For example, compounding interest daily will result in more interest earned than compounding interest monthly.
Q6: How can I calculate compound interest manually?
A6: To calculate compound interest manually, you can use the formula:
A = P(1 + r/n)^(nt)
Where:
- A is the amount of money accumulated after n years, including interest
- P is the principal amount (initial investment)
- r is the annual interest rate (in decimal form)
- n is the number of times that interest is compounded per year
- t is the time the money is invested for in years
Q7: What is the impact of interest rate on compound interest?
A7: The interest rate has a significant impact on compound interest. A higher interest rate will result in more interest earned, while a lower interest rate will result in less interest earned.
Q8: Can compound interest be negative?
A8: Yes, compound interest can be negative. This occurs when the interest rate is negative, or when the principal amount is reduced due to fees or other charges.
Q9: How can I use compound interest to my advantage?
A9: To use compound interest to your advantage, you can:
- Invest in a high-yield savings account or certificate of deposit (CD)
- Take advantage of compound interest by investing for a longer period
- Consider a loan with a low interest rate and compound interest
Q10: What are some common mistakes to avoid when calculating compound interest?
A10: Some common mistakes to avoid when calculating compound interest include:
- Forgetting to account for compounding frequency
- Using the wrong interest rate or principal amount
- Not considering the impact of fees or other charges
Conclusion
In conclusion, compound interest is a powerful tool for calculating the value of an account with interest. By understanding the formula and applying it correctly, we can accurately calculate the value of an account with compound interest. We hope this article has provided a clear explanation of compound interest and its application.
Additional Resources
For more information on compound interest, you can consult the following resources:
Compound Interest Calculator
To calculate compound interest, you can use the following calculator:
| Principal Amount | Interest Rate | Compounding Frequency | Time | Interest Earned |
|---|---|---|---|---|
| $1,000 | 5% | Annually | 5 years | $275.31 |
| $5,000 | 3% | Quarterly | 10 years | $1,632.50 |
| $10,000 | 2% | Monthly | 20 years | $4,321.19 |
Note: The calculator is for illustrative purposes only and should not be used for actual financial calculations.