If $\$550$ Are Deposited Into An Account With A $7.5\%$ Interest Rate, Compounded Monthly, What Is The Balance After 14 Years? Use The Formula: $\[ F = P \left(1 + \frac{r}{n}\right)^{nt} \\]Type Your Answer As A Number,

Feb 28, 2025 by ADMIN 225 views

$If $550 are deposited into an account with a $7.5\%$ interest rate, compounded monthly, what is the balance after 14 years?$

Understanding the Problem

To solve this problem, we need to use the formula for compound interest, which is given by:

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

Where:

$F$ 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 form)
$n$ is the number of times that interest is compounded per year
$t$ is the time the money is invested or borrowed for, in years

Breaking Down the Given Information

In this problem, we are given the following information:

$P = 550$ (the initial deposit)
$r = 7.5\% = 0.075$ (the annual interest rate)
$n = 12$ (since the interest is compounded monthly)
$t = 14$ years (the time the money is invested for)

Applying the Formula

Now that we have all the necessary information, we can plug it into the formula to find the future value of the investment:

${ F = 550 \left(1 + \frac{0.075}{12}\right)^{12 \times 14} }$

Calculating the Future Value

To calculate the future value, we need to follow the order of operations (PEMDAS):

Calculate the value inside the parentheses: $\left(1 + \frac{0.075}{12}\right)$ ${ \left(1 + \frac{0.075}{12}\right) = \left(1 + 0.00625\right) = 1.00625 }$
Raise the result to the power of $nt$ : $\left(1.00625\right)^{12 \times 14}$ ${ \left(1.00625\right)^{12 \times 14} = \left(1.00625\right)^{168} }$
Multiply the result by the principal investment amount: $550 \times \left(1.00625\right)^{168}$

Using a Calculator to Find the Future Value

To find the future value, we can use a calculator to evaluate the expression:

${ 550 \times \left(1.00625\right)^{168} \approx 550 \times 3.352 }$

Calculating the Final Answer

${ 550 \times 3.352 \approx 1841.60 }$

Conclusion

Therefore, the balance after 14 years is approximately $1841.60.

Discussion

This problem demonstrates the power of compound interest in growing an investment over time. By understanding the formula for compound interest and applying it to a given scenario, we can calculate the future value of an investment and make informed decisions about our financial future.

Additional Examples

Here are a few additional examples of how to use the formula for compound interest:

If $1000 is deposited into an account with a $5\%$ interest rate, compounded quarterly, what is the balance after 10 years?
If $500 is deposited into an account with a $10\%$ interest rate, compounded annually, what is the balance after 20 years?
If $2000 is deposited into an account with a $12\%$ interest rate, compounded monthly, what is the balance after 5 years?

Solutions to Additional Examples

If $1000 is deposited into an account with a $5\%$ interest rate, compounded quarterly, what is the balance after 10 years? ${ F = 1000 \left(1 + \frac{0.05}{4}\right)^{4 \times 10} }$ ${ F = 1000 \left(1 + 0.0125\right)^{40} }$ ${ F = 1000 \left(1.0125\right)^{40} }$ ${ F \approx 1000 \times 2.208 }$ ${ F \approx 2208 }$
If $500 is deposited into an account with a $10\%$ interest rate, compounded annually, what is the balance after 20 years? ${ F = 500 \left(1 + 0.10\right)^{20} }$ ${ F = 500 \left(1.10\right)^{20} }$ ${ F \approx 500 \times 6.727 }$ ${ F \approx 3363.50 }$
If $2000 is deposited into an account with a $12\%$ interest rate, compounded monthly, what is the balance after 5 years? ${ F = 2000 \left(1 + \frac{0.12}{12}\right)^{12 \times 5} }$ ${ F = 2000 \left(1 + 0.01\right)^{60} }$ ${ F = 2000 \left(1.01\right)^{60} }$ ${ F \approx 2000 \times 1.822 }$ ${ F \approx 3644 }$

Conclusion

In conclusion, the formula for compound interest is a powerful tool for calculating the future value of an investment. By understanding the formula and applying it to a given scenario, we can make informed decisions about our financial future.

Understanding Compound Interest

Q&A

Q: What is compound interest?

A: Compound interest is the interest earned on both the principal amount and any accrued interest over time. It is a powerful tool for growing an investment over time.

Q: How does compound interest work?

A: Compound interest works by applying the interest rate to the principal amount and any accrued interest over time. This creates a snowball effect, where the interest earned on the principal amount is added to the principal amount, and then the interest rate is applied to the new total.

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

A: The key factors that affect compound interest are:

Principal amount: The initial amount invested
Interest rate: The rate at which interest is earned
Time: The length of time the money is invested
Compounding frequency: The frequency at which interest is compounded (e.g. monthly, quarterly, annually)

Q: How can I calculate compound interest?

A: You can calculate compound interest using the formula:

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

Where:

F is the future value of the investment
P is the principal amount
r is the interest rate
n is the compounding frequency
t is the time the money is invested

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

A: Simple interest is the interest earned only on the principal amount, whereas compound interest is the interest earned on both the principal amount and any accrued interest over time.

Q: How can I use compound interest to grow my investment?

A: You can use compound interest to grow your investment by:

Investing a large principal amount
Choosing a high interest rate
Investing for a long period of time
Compounding interest frequently

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

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

Not understanding the interest rate and compounding frequency
Not considering the time value of money
Not diversifying your investments
Not regularly reviewing and adjusting your investment strategy

Q: How can I use compound interest to achieve my financial goals?

A: You can use compound interest to achieve your financial goals by:

Setting clear financial goals
Creating a comprehensive investment plan
Regularly reviewing and adjusting your investment strategy
Staying disciplined and patient

Conclusion

Compound interest is a powerful tool for growing an investment over time. By understanding the formula for compound interest and applying it to a given scenario, we can make informed decisions about our financial future. By avoiding common mistakes and using compound interest to achieve our financial goals, we can create a secure and prosperous financial future.

Additional Resources

Compound Interest Calculator: A tool for calculating compound interest
Investment Guide: A comprehensive guide to investing and achieving financial goals
Financial Planning: A resource for creating a comprehensive investment plan and achieving financial goals

Frequently Asked Questions

Q: What is the difference between compound interest and simple interest? A: Compound interest is the interest earned on both the principal amount and any accrued interest over time, whereas simple interest is the interest earned only on the principal amount.
Q: How can I calculate compound interest? A: You can calculate compound interest using the formula: F = P (1 + r/n)^(nt)
Q: What are some common mistakes to avoid when using compound interest? A: Some common mistakes to avoid when using compound interest include not understanding the interest rate and compounding frequency, not considering the time value of money, not diversifying your investments, and not regularly reviewing and adjusting your investment strategy.