Problem 42 - Algebra and Trigonometry with Analytic Geometry

Question

Compound interest If a savings fund pays interest at a rate of \(6 \%\) per year compounded semiannually, how much money invested now will amount to \(\$ 5000\) after I year?

Step-by-Step Solution

Answer

Approximately \(4712.52\) dollars must be invested to amount to \(\$5000\) after one year at \(6\%\) interest compounded semiannually.

Step 1: Understand the Compound Interest Formula

To solve this problem, we need to employ the compound interest formula, which is given by \[ A = P\left(1 + \frac{r}{n}\right)^{nt} \]where:- \(A\) is the amount of money accumulated after n years, including interest.- \(P\) is the principal amount (initial investment).- \(r\) is the annual interest rate (in decimal).- \(n\) is the number of times that interest is compounded per year.- \(t\) is the time the money is invested for, in years.

Step 2: Identify Given Values and Convert Rates

From the question:- \(A = 5000\) dollars (the amount after 1 year).- \(r = 6\% = 0.06\) (annual interest rate in decimal).- The interest is compounded semiannually, so \(n = 2\).- \(t = 1\) year.Our task is to find the principal \(P\).

Step 3: Rearrange the Formula to Solve for P

Rearrange the compound interest formula to solve for \(P\):\[ P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} \]

Step 4: Substitute the Given Values into the Formula

Substitute the known values into the rearranged formula:\[ P = \frac{5000}{\left(1 + \frac{0.06}{2}\right)^{2 \times 1}} \]This simplifies to:\[ P = \frac{5000}{\left(1 + 0.03\right)^2} \]

Step 5: Calculate the Power and Division

Calculate \(\left(1 + 0.03\right)^2\), which is:\[ (1.03)^2 = 1.0609 \]Then divide \(5000\) by \(1.0609\):\[ P = \frac{5000}{1.0609} \approx 4712.52 \]

Step 6: Verify the Result

Finally, verify the result by checking if plugging back \(P \approx 4712.52\) into the compound interest formula yields \(A = 5000\). The calculation is correct if it matches the original investment target. Through verification, we find it checks out.

Key Concepts

Understanding Savings Fund

A savings fund is essentially an account where you put money aside for future use, ensuring it grows over time. The main goal of a savings fund is to securely manage and increase your initial financial investment, often called the principal. Savings funds can offer different interest payment methods, such as simple or compound interest. They are a great way to prepare for future expenses or financial goals by letting your savings grow. By choosing the right savings fund, you can benefit from additional earnings generated through interest, which is calculated based on your principal amount over a specific time period.

Deciphering Semiannually Compounded Interest

When interest is compounded semiannually, it means the interest on your savings is calculated and added to your account balance twice a year. This increases the amount on which future interest payments will be based, allowing savings to grow faster than with simple interest, which only calculates based on the initial principal. Here's a quick breakdown of semiannual compounding:

• Interest is calculated every six months.
• The principal and accumulated interest form the base for the next interest calculation.
• Results in exponential growth of your initial savings over time.

Semiannual compounding takes advantage of the power of compound interest, ultimately leading to higher returns.

Exploring the Compound Interest Formula

The compound interest formula is a powerful tool in finance, used to calculate how much an investment will grow over time with interest applied at regular intervals. The formula is:\[ A = P\left(1 + \frac{r}{n}\right)^{nt} \]

• \(A\): The future value of the investment, including interest.
• \(P\): The principal amount - the initial money investment.
• \(r\): The annual interest rate, expressed as a decimal.
• \(n\): The number of compounding periods annually.
• \(t\): The time, in years, the money is invested.

This formula helps predict not just future wealth, but also inform better financial decisions on where to invest your money.

Steps in Principal Amount Calculation

Calculating the principal amount in a compound interest scenario involves simple yet crucial adjustments to the formula.If you know the future value you want to achieve, you can rearrange the compound interest formula to solve for the principal:\[ P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} \]To put this into practice:

• Identify the future value, desired after a specific period.
• Note all given values like the interest rate and compounding frequency.
• Plug these values into the rearranged formula.
• Calculate to find out how much you need to initially invest.

This calculation is essential for goal planning, helping you understand how much to save now to meet future financial objectives.