Faith Plans To Buy A Car For $15,000 In 3 Years. How Much Money Should She Invest If The Bank Is Offering 5% Interest Compounded Continuously? A. $17,427.51 B. $12,910.62 C. $15,769.07 D. $47,307.20
Introduction
Faith is planning to buy a car worth $15,000 in 3 years. To achieve her goal, she needs to invest a certain amount of money in a bank account that offers a 5% interest rate compounded continuously. In this article, we will explore the concept of continuous compounding and calculate the amount of money Faith should invest to reach her goal.
Understanding Continuous Compounding
Continuous compounding is a type of interest calculation where the interest is compounded on an initial principal amount over a period of time. The formula for continuous compounding is:
A = P * e^(rt)
Where:
- A is the amount of money accumulated after n years, including interest
- P is the principal amount (initial investment)
- e is the base of the natural logarithm (approximately 2.71828)
- r is the annual interest rate (in decimal form)
- t is the time the money is invested for, in years
Calculating Faith's Investment
To calculate the amount of money Faith should invest, we need to use the formula for continuous compounding. We know that:
- A = $15,000 (the amount Faith wants to accumulate in 3 years)
- r = 5% = 0.05 (the annual interest rate)
- t = 3 years (the time Faith has to invest her money)
We need to solve for P, the principal amount. Rearranging the formula, we get:
P = A / (e^(rt))
Plugging in the values, we get:
P = $15,000 / (e^(0.05*3)) P = $15,000 / (e^0.15) P = $15,000 / 1.16183 P = $12,910.62
Conclusion
Faith should invest $12,910.62 to accumulate $15,000 in 3 years, assuming a 5% interest rate compounded continuously. This calculation demonstrates the power of continuous compounding, which can help Faith reach her financial goal with a smaller initial investment.
Comparison with Other Options
Let's compare our result with the other options:
- A. $17,427.51: This is the amount of money Faith would need to invest if the interest were compounded annually, not continuously.
- B. $12,910.62: This is the correct answer, as calculated above.
- C. $15,769.07: This is the amount of money Faith would need to invest if the interest were compounded continuously, but with a higher interest rate (6%).
- D. $47,307.20: This is the amount of money Faith would need to invest if the interest were compounded continuously, but with a much higher interest rate (20%).
Real-World Applications
Continuous compounding has many real-world applications, including:
- Savings accounts: Many savings accounts offer interest rates that are compounded continuously, allowing customers to earn more interest over time.
- Investments: Continuous compounding can be used to calculate the future value of investments, such as stocks or bonds.
- Loans: Continuous compounding can be used to calculate the interest on loans, such as mortgages or car loans.
Conclusion
Q&A: Continuous Compounding
Q: What is continuous compounding?
A: Continuous compounding is a type of interest calculation where the interest is compounded on an initial principal amount over a period of time. The formula for continuous compounding is:
A = P * e^(rt)
Where:
- A is the amount of money accumulated after n years, including interest
- P is the principal amount (initial investment)
- e is the base of the natural logarithm (approximately 2.71828)
- r is the annual interest rate (in decimal form)
- t is the time the money is invested for, in years
Q: How does continuous compounding differ from other types of compounding?
A: Continuous compounding differs from other types of compounding, such as annual compounding, in that it takes into account the fact that interest is compounded continuously over time. This means that the interest is compounded not just at the end of each year, but at every instant in time.
Q: What are some real-world applications of continuous compounding?
A: Some real-world applications of continuous compounding include:
- Savings accounts: Many savings accounts offer interest rates that are compounded continuously, allowing customers to earn more interest over time.
- Investments: Continuous compounding can be used to calculate the future value of investments, such as stocks or bonds.
- Loans: Continuous compounding can be used to calculate the interest on loans, such as mortgages or car loans.
Q: How can I calculate the future value of an investment using continuous compounding?
A: To calculate the future value of an investment using continuous compounding, you can use the formula:
A = P * e^(rt)
Where:
- A is the amount of money accumulated after n years, including interest
- P is the principal amount (initial investment)
- e is the base of the natural logarithm (approximately 2.71828)
- r is the annual interest rate (in decimal form)
- t is the time the money is invested for, in years
Q: What is the formula for continuous compounding in terms of the number of periods?
A: The formula for continuous compounding in terms of the number of periods 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)
- e is the base of the natural logarithm (approximately 2.71828)
- r is the annual interest rate (in decimal form)
- n is the number of times interest is compounded per year
- t is the time the money is invested for, in years
Q: How can I use continuous compounding to calculate the interest on a loan?
A: To calculate the interest on a loan using continuous compounding, you can use the formula:
A = P * e^(rt)
Where:
- A is the amount of money accumulated after n years, including interest
- P is the principal amount (initial investment)
- e is the base of the natural logarithm (approximately 2.71828)
- r is the annual interest rate (in decimal form)
- t is the time the money is borrowed for, in years
Q: What are some common mistakes to avoid when using continuous compounding?
A: Some common mistakes to avoid when using continuous compounding include:
- Not taking into account the fact that interest is compounded continuously over time
- Not using the correct formula for continuous compounding
- Not using the correct values for the variables in the formula
Conclusion
In conclusion, continuous compounding is a powerful tool for calculating the future value of investments and loans. By understanding the formula for continuous compounding and avoiding common mistakes, you can make informed decisions about your financial goals and investments.