An Account With An Initial Balance Of $800 Earns 4.5% Interest For 3 Years. What Is The Ending Balance? Give Your Answer To The Nearest Cent.
Understanding the Problem
To calculate the ending balance of an account with an initial balance of $800 earning 4.5% interest for 3 years, we need to use the concept of compound interest. Compound interest is the interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods on a deposit or loan. In this case, the interest is compounded annually, meaning it is calculated and added to the principal once per year.
Formula for Compound Interest
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 balance).
- 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.
Applying the Formula to the Problem
In this problem, we have:
- P = $800 (initial balance)
- r = 4.5% or 0.045 (annual interest rate in decimal form)
- n = 1 (compounded annually)
- t = 3 years
We can now plug these values into the formula:
A = 800(1 + 0.045/1)^(1*3) A = 800(1 + 0.045)^3 A = 800(1.045)^3
Calculating the Ending Balance
To calculate the ending balance, we need to calculate the value of (1.045)^3 and then multiply it by the initial balance of $800.
(1.045)^3 ≈ 1.144 A = 800 * 1.144 A ≈ 914.72
Rounding to the Nearest Cent
The ending balance is approximately $914.72. Since we are asked to give the answer to the nearest cent, the ending balance is $914.72.
Conclusion
In this problem, we used the formula for compound interest to calculate the ending balance of an account with an initial balance of $800 earning 4.5% interest for 3 years. We found that the ending balance is approximately $914.72.
Understanding the Impact of Time and Interest Rate
The time period and interest rate have a significant impact on the ending balance. In this case, the interest rate of 4.5% per year and the time period of 3 years resulted in an ending balance of $914.72. If the interest rate or time period were to change, the ending balance would also change.
Calculating the Interest Earned
To calculate the interest earned, we can subtract the initial balance from the ending balance:
Interest Earned = Ending Balance - Initial Balance Interest Earned = $914.72 - $800 Interest Earned = $114.72
Conclusion
In this problem, we calculated the ending balance of an account with an initial balance of $800 earning 4.5% interest for 3 years. We found that the ending balance is approximately $914.72. We also calculated the interest earned, which is $114.72.
Real-World Applications
The concept of compound interest is used in many real-world applications, such as:
- Savings accounts
- Certificates of deposit (CDs)
- Bonds
- Loans
- Investments
Understanding compound interest is essential for making informed decisions about personal finance and investments.
Final Thoughts
In conclusion, the ending balance of an account with an initial balance of $800 earning 4.5% interest for 3 years is approximately $914.72. The time period and interest rate have a significant impact on the ending balance, and understanding compound interest is essential for making informed decisions about personal finance and investments.
Q: What is compound interest?
A: Compound interest is the interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods on a deposit or loan. In other words, it's the interest earned on both the principal and any accrued interest.
Q: How does compound interest work?
A: Compound interest works by adding the interest earned in a period to the principal, so that the interest earned in the next period is calculated on the new, higher balance. This process is repeated for each period, resulting in a snowball effect that can lead to significant growth 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 earned, expressed as a percentage.
- Time: The length of time the money is invested or borrowed for.
- Compounding frequency: The number of times interest is compounded per year.
Q: How often is interest compounded?
A: Interest can be compounded annually, semi-annually, quarterly, or monthly, depending on the type of account or loan. The more frequently interest is compounded, the faster the balance will grow.
Q: What is the difference between simple interest and compound interest?
A: Simple interest is calculated only on the principal, whereas compound interest is calculated on both the principal and any accrued interest. This means that compound interest grows faster than simple interest over time.
Q: Can compound interest be negative?
A: Yes, compound interest can be negative if the interest rate is negative or if the principal is reduced by fees or withdrawals. This is known as negative compounding.
Q: How can I maximize compound interest?
A: To maximize compound interest, you can:
- Invest for a longer period: The longer you invest, the more time your money has to grow.
- Choose a higher interest rate: A higher interest rate will result in more interest earned over time.
- Compound interest more frequently: Compounding interest more frequently will result in faster growth.
- Avoid fees and withdrawals: Fees and withdrawals can reduce the principal and interest earned, slowing down growth.
Q: What are some real-world examples of compound interest?
A: Compound interest is used in many real-world applications, such as:
- Savings accounts: Many savings accounts offer compound interest, allowing your money to grow over time.
- Certificates of deposit (CDs): CDs are time deposits offered by banks with a fixed interest rate and maturity date.
- Bonds: Bonds are debt securities issued by companies or governments to raise capital.
- Loans: Compound interest is used to calculate interest on loans, such as mortgages and credit cards.
Q: Can I use compound interest to calculate interest on a loan?
A: Yes, compound interest can be used to calculate interest on a loan. However, you will need to use a formula that takes into account the loan's principal, interest rate, and compounding frequency.
Q: How can I calculate compound interest manually?
A: 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 balance).
- 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.
You can also use a compound interest calculator or spreadsheet to make the calculation easier.
Q: What are some common mistakes to avoid when using compound interest?
A: Some common mistakes to avoid when using compound interest include:
- Not considering compounding frequency: Failing to consider the compounding frequency can result in inaccurate calculations.
- Not accounting for fees and withdrawals: Failing to account for fees and withdrawals can reduce the principal and interest earned, slowing down growth.
- Not using the correct interest rate: Using the wrong interest rate can result in inaccurate calculations.
- Not considering the time value of money: Failing to consider the time value of money can result in inaccurate calculations.
By understanding compound interest and avoiding these common mistakes, you can make informed decisions about your finances and investments.