What Amount, { A $}$, Will Our Account Have After 5 Years If It Earns An Annual Rate Of { 9% $}$ Compounded Quarterly, Starting With { $450$} ? R O U N D T O T H E N E A R E S T C E N T . ? Round To The Nearest Cent. ? R O U N D T O T H E N E A Res T Ce N T . {$ A = $ $}$ { \square$}$

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What Amount Will Our Account Have After 5 Years?

Understanding the Problem

We are given an initial principal amount of $450, an annual interest rate of 9%, and a compounding frequency of quarterly. The goal is to determine the future value of the account after 5 years, rounded to the nearest cent.

The Formula for Compound Interest

The formula for compound interest is:

A = P (1 + r/n)^(nt)

Where:

  • A is the future value of the account
  • P is the principal amount (initial deposit)
  • r is the annual interest rate (in decimal form)
  • n is the number of times the interest is compounded per year
  • t is the time the money is invested for, in years

Breaking Down the Given Values

  • P = $450 (initial deposit)
  • r = 9% = 0.09 (annual interest rate in decimal form)
  • n = 4 (quarterly compounding)
  • t = 5 years

Plugging in the Values

Now, let's substitute the given values into the compound interest formula:

A = 450 (1 + 0.09/4)^(4*5)

Simplifying the Expression

To simplify the expression, we can first calculate the value inside the parentheses:

1 + 0.09/4 = 1 + 0.0225 = 1.0225

Now, we can raise this value to the power of 20 (since 4*5 = 20):

A = 450 (1.0225)^20

Calculating the Future Value

Using a calculator or a computer program, we can calculate the future value of the account:

A ≈ $714.19

Rounding to the Nearest Cent

Rounding $714.19 to the nearest cent gives us a final answer of:

$714.19

Conclusion

In this problem, we used the compound interest formula to determine the future value of an account after 5 years, given an initial principal amount of $450, an annual interest rate of 9%, and a compounding frequency of quarterly. The final answer is $714.19, rounded to the nearest cent.

Additional Information

  • The formula for compound interest can be used to calculate the future value of an account with any combination of principal amount, interest rate, compounding frequency, and time period.
  • The formula assumes that the interest rate remains constant over the entire time period.
  • The formula can be used to calculate the future value of an account with any type of compounding, including monthly, quarterly, or annually.

Real-World Applications

  • Compound interest is used in a variety of real-world applications, including:
    • Savings accounts
    • Certificates of deposit (CDs)
    • Bonds
    • Investments
    • Retirement accounts
  • Understanding compound interest can help individuals make informed decisions about their financial investments and savings.

Common Mistakes

  • Failing to account for compounding frequency
  • Using an incorrect interest rate or principal amount
  • Not rounding the final answer to the nearest cent
  • Not considering the time value of money

Tips and Tricks

  • Use a calculator or computer program to simplify complex calculations
  • Break down the problem into smaller, more manageable parts
  • Double-check the values and calculations to ensure accuracy
  • Consider using a financial calculator or spreadsheet to calculate compound interest

Conclusion

In conclusion, the compound interest formula is a powerful tool for calculating the future value of an account. By understanding the formula and its applications, individuals can make informed decisions about their financial investments and savings.
Compound Interest Q&A

Frequently Asked Questions About Compound Interest

Q: What is compound interest?

A: Compound interest is the interest earned on both the principal amount and any accrued interest over time. It's a powerful force that can help your savings grow exponentially.

Q: How does compound interest work?

A: Compound interest works by adding the interest earned on the principal amount to the principal amount, and then calculating the interest on the new balance. This process is repeated over time, resulting in a snowball effect that can lead to significant growth.

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

A: The key factors that affect compound interest are:

  • Principal amount: The initial amount of money invested
  • Interest rate: The rate at which interest is earned
  • Compounding frequency: The number of times interest is compounded per year
  • Time period: The length of time the money is invested for

Q: What is the formula for compound interest?

A: The formula for compound interest is:

A = P (1 + r/n)^(nt)

Where:

  • A is the future value of the account
  • P is the principal amount
  • r is the annual interest rate
  • n is the number of times interest is compounded per year
  • t is the time period

Q: How often is interest compounded?

A: Interest can be compounded daily, monthly, quarterly, or annually, depending on the type of account and the financial institution.

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

A: Simple interest is calculated only on the principal amount, while compound interest is calculated on both the principal amount and any accrued interest.

Q: Can compound interest be negative?

A: Yes, compound interest can be negative if the interest rate is negative or if the principal amount is reduced over time.

Q: How can I maximize my compound interest?

A: To maximize your 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 can lead to faster growth.
  • Compound interest more frequently: Compounding interest more frequently can lead to faster growth.
  • Avoid withdrawals: Avoid withdrawing money from your account to allow it to grow.

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

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

  • Failing to account for compounding frequency
  • Using an incorrect interest rate or principal amount
  • Not rounding the final answer to the nearest cent
  • Not considering the time value of money

Q: Can I use a calculator or computer program to calculate compound interest?

A: Yes, you can use a calculator or computer program to calculate compound interest. Many financial calculators and spreadsheet programs have built-in functions for calculating compound interest.

Q: What are some real-world applications of compound interest?

A: Compound interest is used in a variety of real-world applications, including:

  • Savings accounts
  • Certificates of deposit (CDs)
  • Bonds
  • Investments
  • Retirement accounts

Q: How can I use compound interest to my advantage?

A: To use compound interest to your advantage, you can:

  • Invest for the long term: Compound interest can help your savings grow exponentially over time.
  • Choose a high-interest rate: A higher interest rate can lead to faster growth.
  • Compound interest frequently: Compounding interest more frequently can lead to faster growth.
  • Avoid withdrawals: Avoid withdrawing money from your account to allow it to grow.

Conclusion

Compound interest is a powerful force that can help your savings grow exponentially over time. By understanding the formula and its applications, you can make informed decisions about your financial investments and savings.