Problem 66 - Contemporary Precalculus

Question

Deal with the compound interest formula \(A=P(1+r)^{t},\) which was discussed in Special Topics \(5.2.A\). At what rate of interest (compounded annually) should you invest 500 dollars if you want to have 1500 dollars in 12 years?

Step-by-Step Solution

Answer

Answer: Approximately 9.5% annually.

Step 1: Write down the known values

We know the initial principal amount (\(P\)) is 500 dollars, the final amount (\(A\)) desired is 1500 dollars and the time period (\(t\)) is 12 years.

Step 2: Substitute the known values into the compound interest formula

Now, replace the values of \(P\), \(A\) and \(t\) in the compound interest formula: \(A = P(1+r)^{t}\) \(1500 = 500(1+r)^{12}\)

Step 3: Solve for the interest rate \(r\)

To solve for \(r\), we will first divide both sides by 500: \(3 = (1+r)^{12}\) Now, take the 12th root on both sides to get: \((3)^{\frac{1}{12}} = 1 + r\) Next, subtract 1 from both sides to find the value of \(r\): \(r = (3)^{\frac{1}{12}} - 1\)

Step 4: Calculate the value of \(r\)

Use a calculator to find the value of \(r\): \(r \approx 0.095\)

Step 5: Convert the value of \(r\) into a percentage

To express the interest rate as a percentage, multiply the value of \(r\) by 100: \(0.095 \times 100 \approx 9.5\%\) Hence, to achieve the goal of having 1500 dollars in 12 years, one should invest 500 dollars at an interest rate of approximately 9.5% compounded annually.

Key Concepts

Interest Rate Calculation

Interest rate calculation is a fundamental part of financial mathematics.
It helps us understand how money grows over time when invested or borrowed.
In the context of compound interest, the interest rate represents how much the initial amount, or principal, increases over each period.
When tackling problems in compound interest, it's essential to determine the correct rate.
In our scenario, this involves deducing the annual interest rate needed for an initial investment, over a set number of years, to grow to a specific amount.
The key steps include:

• Identifying the known variables: principal, future amount, and time period.
• Using these to express and manipulate the compound interest formula accordingly.
• Solving the equation for the rate, often requiring algebraic rearrangement and evaluation of roots, as shown in our example where we solved for the 12th root.

Calculating the correct interest rate is crucial for making informed financial decisions, ensuring that goals are met within the desired timeline.
Once the interest rate is found, it can be converted into a percentage to offer a clearer understanding of the annual growth rate required.

Compound Interest Formula

The compound interest formula is a pivotal tool in financial mathematics.
It helps in understanding how investments grow over time.
This formula considers the rate at which interest is applied to both the original principal and previously earned interest.The general formula is:\[ A = P(1+r)^{t} \]Where:

• \( A \) is the amount of money accumulated after the investment period, inclusive of interest.
• \( P \) is the principal amount, or the initial sum of money invested.
• \( r \) is the annual interest rate (expressed as a decimal).
• \( t \) is the time the money is invested for, in years.

The formula's power lies in its ability to calculate the future value of any investment considering the compounding effect.
Compounding means that interest in each future period is earned not just on the original principal, but also on accumulated interest from previous periods.
It's a crucial concept for understanding how investments can grow exponentially over time and is regularly employed in personal finance, banking, and investment sectors.

Financial Mathematics

Financial mathematics is a branch of applied mathematics that deals with financial markets.
It encompasses various calculations and formulas, including those for compound interest, which aid individuals and businesses in making sound financial decisions.
Key aspects of financial mathematics include:

• Evaluating investment options to maximize returns, such as using the compound interest formula to predict future earnings.
• Understanding the time value of money, which suggests that money available today is worth more than the same amount in the future due to its earning potential.
• Performing calculations like present value, future value, and interest rate determinations, which are fundamental in planning for financial growth and security.

By mastering financial mathematics concepts, individuals can make informed choices about loans, investments, savings, retirement planning, and more.
These calculations ensure that both personal and organizational financial goals meet realistic and strategic benchmarks.