Problem 80 - Algebra and Trigonometry

Question

A sum of \(\$ 1000\) was invested for 4 years, and the interest was compounded semiannually. If this sum amounted to \(\$ 1435.77\) in the given time, what was the interest rate?

Step-by-Step Solution

Answer

The interest rate was approximately 9.07% per annum.

Step 1: Understand the Formula for Compound Interest

Compound interest is calculated using the formula \( A = P \left(1 + \frac{r}{n}\right)^{nt} \), where \(A\) is the amount of money accumulated after \(n\) periods, \(P\) is the principal amount (initial investment), \(r\) is the annual interest rate (as a decimal), \(n\) is the number of times that interest is compounded per year, and \(t\) is the time the money is invested or borrowed for, in years.

Step 2: Identify the Known Values

From the exercise, we know: - \( P = 1000 \)- \( A = 1435.77 \)- \( n = 2 \) (since the interest is compounded semiannually)- \( t = 4 \) years.

Step 3: Plug Known Values into the Equation

Substitute the known values into the formula: \[ 1435.77 = 1000 \left(1 + \frac{r}{2}\right)^{8} \] This simplifies to: \[ 1435.77 = 1000 \left(1 + \frac{r}{2}\right)^{8} \]

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

First, divide both sides of the equation by 1000: \[ 1.43577 = \left(1 + \frac{r}{2}\right)^{8} \] To isolate \(r\), first take the eighth root of both sides: \[ 1.43577^{\frac{1}{8}} = 1 + \frac{r}{2} \] Calculate the eighth root: \[ 1.04535 \approx 1 + \frac{r}{2} \] Subtract 1 from both sides: \[ 0.04535 = \frac{r}{2} \] Finally, multiply by 2 to solve for \(r\): \[ r = 0.0907 \]

Step 5: Convert the Decimal to a Percentage

The interest rate \(r\) as a decimal is 0.0907. To convert this into a percentage, multiply by 100: \[ r = 9.07\% \]

Key Concepts

Interest Rate Calculation

Calculating the interest rate is a key part of determining how much your investment will grow over time. In the context of compound interest, the formula to find the final amount is essential. Here's a simplified approach to help you grasp the calculation:

The basic formula for compound interest is:

• \( A = P \left(1 + \frac{r}{n}\right)^{nt} \)

• **A**: the amount of money accumulated after n years, including interest.
• **P**: the principal amount, which is the initial amount of money invested.
• **r**: the annual interest rate (expressed as a decimal).
• **n**: the number of times that interest is compounded per year.
• **t**: the number of years the money is invested for.

In our example, we're given all other values and asked to find the unknown interest rate. With figures plugged into the formula, isolate **r** using algebra to discover the interest rate that turns \\(1000 into \\)1435.77 over four years of semiannual compounding. Understanding each component of this formula will allow you to calculate interest rates with confidence, just like a financial expert!

Investment Growth

Investment growth is all about seeing how your initial deposit evolves over time in a savings account or another financial product, thanks to interest. Compound interest specifically means you're earning "interest on interest." This magical quality results in faster and larger growth of your investment than simple interest, which only pays interest on the original sum.

Consider our initial investment of \\(1000. Over four years, it grew to \\)1435.77 with interest compounding twice a year. This demonstrates compound interest at work. Each period, not only the principal is earning interest, but also any previously earned interest is doing so as well.

Imagine planting a tree; each year, it adds rings, representing growth. With compound interest, those yearly rings get thicker as both the base and previous rings grow. This allows for exponential growth over time.

Remember, the key factors influencing your investment growth include:

• How often the interest compounds.
• The duration for which funds are invested.
• The annual interest rate.

By understanding these concepts, you can better optimize your savings strategy to maximize returns.

Mathematical Formulas

Mathematical formulas, like the compound interest formula, are crucial in finance. They allow us not only to predict future values but also to solve for unknowns, like the interest rate.

Solving such problems involves a logical sequence of algebraic steps. For example, in our exercise, once we have\[1435.77 = 1000 \left(1 + \frac{r}{2}\right)^8 \]you first divide by 1000 to simplify:\[1.43577 = \left(1 + \frac{r}{2}\right)^8 \]To further isolate \( r \), you take the eighth root:\[1.43577^{\frac{1}{8}} = 1 + \frac{r}{2} \]Each step reduces the complexity and isolates the unknown value, \( r \). By mastering these steps, you can tackle any compound interest calculations. The beauty of mathematical formulas lies in their power to demystify complex finance concepts. This makes them invaluable tools whether calculating returns on investment or balancing your checkbook.