Select The Correct Answer.Jenny Borrowed $$ 500$ For Five Years At 4 Percent Interest, Compounded Annually. What Is The Total Amount She Will Have Paid When She Pays Off The Loan?Total Amount = = = P(1 + R)^t$$A. $$
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. In the case of Jenny's loan, the interest is compounded annually, meaning it is calculated and added to the principal once per year. The formula for calculating the total amount paid on a loan with compound interest is given by:
Total Amount = P(1 + r)^t
Where:
- P is the principal amount (the initial amount borrowed)
- r is the annual interest rate (expressed as a decimal)
- t is the number of years the money is borrowed for
Calculating the Total Amount Paid
In Jenny's case, the principal amount (P) is $500, the annual interest rate (r) is 4% or 0.04 (expressed as a decimal), and the loan is for five years (t = 5). Plugging these values into the formula, we get:
Total Amount = 500(1 + 0.04)^5
Total Amount = 500(1.04)^5
Total Amount = 500(1.21664736)
Total Amount ≈ 608.32
Therefore, the total amount Jenny will have paid when she pays off the loan is approximately $608.32.
Breaking Down the Calculation
To understand how the calculation works, let's break it down step by step:
- Year 1: The interest for the first year is 4% of $500, which is $20. The total amount after the first year is $500 + $20 = $520.
- Year 2: The interest for the second year is 4% of $520, which is $20.80. The total amount after the second year is $520 + $20.80 = $540.80.
- Year 3: The interest for the third year is 4% of $540.80, which is $21.63. The total amount after the third year is $540.80 + $21.63 = $562.43.
- Year 4: The interest for the fourth year is 4% of $562.43, which is $22.50. The total amount after the fourth year is $562.43 + $22.50 = $584.93.
- Year 5: The interest for the fifth year is 4% of $584.93, which is $23.40. The total amount after the fifth year is $584.93 + $23.40 = $608.33.
As we can see, the total amount paid after five years is approximately $608.33, which is very close to the calculated value of $608.32.
Conclusion
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. It is a powerful force that can add up over time, making it an essential concept to understand when it comes to personal finance.
Q: How does compound interest work?
A: Compound interest works by calculating interest on both the principal amount and any accrued interest. This means that the interest is added to the principal, and then the interest is calculated on the new total. For example, if you borrow $100 at a 10% annual interest rate, the interest for the first year would be $10, making the total $110. In the second year, the interest would be calculated on the new total of $110, resulting in $11 in interest, making the total $121.
Q: What is the formula for calculating compound interest?
A: The formula for calculating compound interest is:
Total Amount = P(1 + r)^t
Where:
- P is the principal amount (the initial amount borrowed)
- r is the annual interest rate (expressed as a decimal)
- t is the number of years the money is borrowed for
Q: How can I calculate compound interest manually?
A: To calculate compound interest manually, you can use the formula above and plug in the values for the principal amount, interest rate, and time period. For example, if you borrow $500 at a 4% annual interest rate for 5 years, you would calculate the total amount as follows:
Total Amount = 500(1 + 0.04)^5
Total Amount = 500(1.04)^5
Total Amount = 500(1.21664736)
Total Amount ≈ 608.32
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 any accrued interest. This means that compound interest can add up much faster than simple interest over time.
Q: How can I use compound interest to my advantage?
A: There are several ways to use compound interest to your advantage:
- Save for the future: By saving money and earning compound interest, you can build up a significant amount of wealth over time.
- Invest in a high-yield savings account: High-yield savings accounts often offer higher interest rates than traditional savings accounts, making them a great option for earning compound interest.
- Take advantage of tax-advantaged accounts: Accounts such as 401(k) and IRA plans offer tax benefits that can help your money grow faster through compound interest.
Q: What are some common mistakes to avoid when dealing with compound interest?
A: Some common mistakes to avoid when dealing with compound interest include:
- Not understanding the interest rate: Make sure you understand the interest rate and how it will affect your savings or debt.
- Not considering compounding frequency: Compound interest can be calculated daily, monthly, or annually, depending on the account. Make sure you understand how often interest is compounded.
- Not taking advantage of tax benefits: Tax-advantaged accounts can help your money grow faster through compound interest. Make sure you take advantage of these benefits.
Conclusion
In conclusion, compound interest is a powerful force that can add up over time. By understanding how it works and taking advantage of it, you can build up a significant amount of wealth over time. Remember to avoid common mistakes and take advantage of tax benefits to make the most of compound interest.