Mary Gets A Loan For $35,000 With A Compounded Interest Rate Of 5.6% Compounded Quarterly For Four Years. How Much Will She Pay Back In Total? Show Your Work For Credit. Just Writing The Answer Will Only Get You 10 Points.

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Understanding the Problem

Mary has taken a loan of $35,000 with a compounded interest rate of 5.6% per annum, compounded quarterly, for a period of four years. To calculate the total amount she will have to repay, we need to use the formula for compound interest:

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

Where:

  • A = the total amount to be repaid
  • P = the principal amount (initial loan amount) = $35,000
  • r = the annual interest rate = 5.6% = 0.056
  • n = the number of times interest is compounded per year = 4 (quarterly)
  • t = the time the money is invested for, in years = 4

Step 1: Convert the Annual Interest Rate to a Quarterly Rate

Since the interest is compounded quarterly, we need to convert the annual interest rate to a quarterly rate. We can do this by dividing the annual interest rate by the number of times interest is compounded per year:

Quarterly interest rate = 5.6%/4 = 1.4%/quarter = 0.014

Step 2: Calculate the Total Amount to be Repaid

Now that we have the quarterly interest rate, we can plug in the values into the compound interest formula:

A = 35000(1 + 0.014)^(4*4) A = 35000(1.014)^16 A = 35000 * 1.257 A = 44195

Calculating the Total Amount to be Repaid with Step-by-Step Calculations

To get full credit, we need to show our work step-by-step. Here are the calculations:

  1. Calculate the total number of periods (nt): nt = 4 * 4 = 16

  2. Calculate the total amount to be repaid using the compound interest formula: A = 35000(1 + 0.014)^(16) A = 35000(1.014)^16

  3. Calculate the value of (1.014)^16: (1.014)^16 ≈ 1.257

  4. Multiply the principal amount by the value calculated in step 3: A = 35000 * 1.257 A ≈ 44195

Conclusion

Based on the calculations above, Mary will have to repay a total of $44,195 in four years, assuming a compounded interest rate of 5.6% per annum, compounded quarterly.

Key Takeaways

  • The formula for compound interest is A = P(1 + r/n)^(nt)
  • To calculate the total amount to be repaid, we need to convert the annual interest rate to a quarterly rate
  • We can use a calculator or perform the calculations step-by-step to get the total amount to be repaid

Real-World Applications

Understanding compound interest is crucial in real-world applications such as:

  • Personal finance: When taking out a loan or investing money, it's essential to understand how compound interest works to make informed decisions.
  • Business: Companies use compound interest to calculate interest on loans and investments, and to determine the total amount to be repaid.
  • Economics: Compound interest plays a significant role in economic growth and development, as it affects the amount of money available for investment and consumption.

Common Mistakes to Avoid

When calculating compound interest, it's essential to avoid the following common mistakes:

  • Not converting the annual interest rate to a quarterly rate
  • Not using the correct formula for compound interest
  • Not performing the calculations step-by-step
  • Not considering the time value of money

Conclusion

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 type of interest that is calculated on a regular basis, such as monthly or quarterly, and is added to the principal amount.

Q: How does compound interest work?

A: Compound interest works by applying the interest rate to the principal amount and any accrued interest over a specific period of time. This process is repeated at regular intervals, such as monthly or quarterly, and the interest is added to the principal amount.

Q: What are the benefits of compound interest?

A: The benefits of compound interest include:

  • Increased returns: Compound interest can lead to higher returns on investments over time.
  • Reduced risk: Compound interest can help reduce the risk of investment by spreading the returns over a longer period of time.
  • Increased wealth: Compound interest can help individuals build wealth over time by earning interest on their investments.

Q: What are the drawbacks of compound interest?

A: The drawbacks of compound interest include:

  • Higher interest rates: Compound interest can lead to higher interest rates, which can be a burden for individuals with high-interest loans or credit card debt.
  • Increased complexity: Compound interest can be complex to understand and calculate, which can lead to errors and misunderstandings.
  • Risk of over-investment: Compound interest can lead to over-investment, which can result in a loss of principal if the investment fails.

Q: How can I calculate compound interest?

A: You can calculate compound interest using the following formula:

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

Where:

  • A = the total amount to be repaid
  • P = the principal amount (initial loan amount)
  • r = the annual interest rate
  • n = the number of times interest is compounded per year
  • t = the time the money is invested for, in years

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

A: The main difference between simple interest and compound interest is that simple interest is calculated only on the principal amount, while compound interest is calculated on both the principal amount and any accrued interest.

Q: How can I avoid common mistakes when calculating compound interest?

A: To avoid common mistakes when calculating compound interest, make sure to:

  • Use the correct formula: Use the compound interest formula to calculate the total amount to be repaid.
  • Convert the annual interest rate to a quarterly rate: Convert the annual interest rate to a quarterly rate to ensure accurate calculations.
  • Perform calculations step-by-step: Perform calculations step-by-step to ensure accuracy and avoid errors.
  • Consider the time value of money: Consider the time value of money when calculating compound interest to ensure accurate results.

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

A: Some real-world applications of compound interest include:

  • Personal finance: Compound interest is used to calculate interest on loans and investments, and to determine the total amount to be repaid.
  • Business: Companies use compound interest to calculate interest on loans and investments, and to determine the total amount to be repaid.
  • Economics: Compound interest plays a significant role in economic growth and development, as it affects the amount of money available for investment and consumption.

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

A: To use compound interest to your advantage, make sure to:

  • Understand the compound interest formula: Understand the compound interest formula to calculate the total amount to be repaid.
  • Choose the right investment: Choose the right investment to earn compound interest, such as a high-yield savings account or a certificate of deposit (CD).
  • Monitor and adjust: Monitor and adjust your investment regularly to ensure you're earning the highest returns possible.

Conclusion

In conclusion, compound interest is a powerful tool that can help individuals build wealth over time. By understanding the compound interest formula and avoiding common mistakes, individuals can use compound interest to their advantage and achieve their financial goals.