Problem 66 - College Algebra

Question

Kyoko has \(\$ 10,000\) that she wants to invest. Her bank has several investment accounts to choose from, all compounding daily. Her goal is to have \(\$ 15,000\) by the time she finishes graduate school in 6 years. To the nearest hundredth of a percent, what should her minimum annual interest rate be in order to reach her goal? (Hint: solve the compound interest formula for the interest rate.)

Step-by-Step Solution

Answer

Kyoko needs a minimum annual interest rate of about 6.68% compounded daily.

Step 1: Understand the Problem

Kyoko wants to invest $10,000 to accumulate $15,000 in 6 years using a daily compounding interest account. The task is to determine the minimum annual interest rate needed to achieve this goal.

Step 2: Use the Compound Interest Formula

The formula for compound interest is \( A = P \left(1 + \frac{r}{n}\right)^{nt} \), where \( A \) is the amount of money accumulated after \( t \) years, including interest, \( P \) is the principal amount (initial investment), \( r \) is the annual interest rate (decimal), \( n \) is the number of times interest is compounded per year, and \( t \) is the number of years the money is invested for.

Step 3: Substitute Values into the Formula

Plug the known values into the compound interest formula: \( A = 15000 \), \( P = 10000 \), \( n = 365 \) (since interest is compounded daily), and \( t = 6 \). The equation becomes \( 15000 = 10000 \left(1 + \frac{r}{365}\right)^{365 \times 6} \).

Step 4: Solve for the Interest Rate \( r \)

Rearrange the equation to solve for \( r \):\[ 1 + \frac{r}{365} = \left(\frac{15000}{10000}\right)^{\frac{1}{2190}} \]Calculate \( \frac{15000}{10000} = 1.5 \). Therefore,\[ 1 + \frac{r}{365} = 1.5^{\frac{1}{2190}} \]

Step 5: Compute the Value of \( r \)

Calculate the right-hand side expression:\( 1.5^{\frac{1}{2190}} \approx 1.000183038 \).So,\[1 + \frac{r}{365} \approx 1.000183038\]Solving for \( r \):\[\frac{r}{365} = 1.000183038 - 1 \r = 365 \times 0.000183038 \r \approx 0.06683687\]

Step 6: Convert the Interest Rate to a Percentage

Convert the calculated interest rate into a percentage by multiplying by 100:\( r \approx 0.06683687 \times 100 \approx 6.68\% \).

Step 7: Conclusion

The minimum annual interest rate Kyoko needs is approximately 6.68% compounded daily in order to reach her goal of $15,000 in 6 years.

Key Concepts

Annual Interest Rate

The annual interest rate is a crucial figure to determine when considering investments, especially when planning for long-term goals. It represents the percentage of return expected on an investment over a year.
For Kyoko, knowing the annual interest rate helps her foresee how much her savings will grow each year while in graduate school.
In financial formulas, it is often denoted as \(r\) and is presented as a decimal when used in calculations. For example, an annual interest rate of 6.68% would be used as 0.0668 in computations.

• The annual interest rate is vital for predicting investment growth.
• It helps compare different investment opportunities.
• Understanding this rate aids in setting realistic financial goals.

Knowing your required annual interest rate can help you plan effectively and choose the right investment account that aligns with your objectives.

Investment Accounts

Investment accounts are financial products designed to help individuals grow their money over time. They vary in terms, rates, and frequency of compounding.
For Kyoko, choosing an account with daily compounding benefits her more, as it accumulates interest more frequently, allowing her initial deposit to grow faster.
An investment account should align with your financial goals, risk tolerance, and the time frame you have for letting your investment grow.

• They provide a platform to earn returns on saved money.
• Different accounts have varying minimum balances, fees, and interest structures.
• Your choice should depend on how often interest is compounded.

Understanding the nature of investment accounts can help make informed decisions that fit into one's financial planning for big milestones like graduate school savings.

Daily Compounding

Daily compounding refers to how often interest is calculated and added to the principal balance of an investment. In Kyoko's case, the interest on her $10,000 investment is compounded daily, which means that calculated interest is added to her balance every single day.
This method of compounding allows her to earn interest on interest previously added, thus accelerating the growth of her investment as compared to less frequent compounding methods like monthly or annually.

• Daily compounding leads to faster accumulation of wealth.
• The formula used is \( A = P \left(1 + \frac{r}{n}\right)^{nt} \) where \( n = 365 \) for daily compounding.
• Interest accrued daily means increased potential over time.

Daily compounding is particularly effective for maximizing returns when saving for significant goals, such as graduate school savings, because every day counts towards total growth.

Graduate School Savings

Graduate school savings involve planning and setting aside funds to cover expenses related to further education, such as tuition, books, and living expenses.
Kyoko's goal to have $15,000 for this purpose after six years means she needs to take a strategic approach to saving and investing. Managing these savings with optimal growth strategies, like investing with the right annual interest rate and using accounts with favorable terms like daily compounding, will help meet future financial needs.

• Savings should account for projected tuition, living expenses, and unforeseen costs.
• Early and consistent saving reduces financial stress during graduate school.
• Strategic planning involves setting realistic goals and choosing appropriate investment vehicles.

Ensuring adequate graduate school savings can significantly impact one's academic focus and stress levels, allowing students to concentrate more on their studies.