Problem 80 - Precalculus Mathematics for Calculus

Question

Interest Rate \(A\) 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 is 9%.

Step 1: Identify the Compound Interest Formula

To find the interest rate, we use the compound interest formula: \( A = P (1 + \frac{r}{n})^{nt} \). Here, \( A \) is the final amount, \( P \) is the principal, \( r \) is the annual interest rate, \( n \) is the number of times interest is compounded per year, and \( t \) is the time in years.

Step 2: Plug in Known Values

We know \( A = 1435.77 \), \( P = 1000 \), \( n = 2 \) (since it's compounded semiannually), and \( t = 4 \). Substitute these values into the formula: \( 1435.77 = 1000 (1 + \frac{r}{2})^{8} \).

Step 3: Isolate the Compound Growth Factor

Divide both sides by 1000 to get the growth factor alone: \( 1.43577 = (1 + \frac{r}{2})^{8} \).

Step 4: Solve for the Growth Factor Base

Take the eighth root to solve for the base of the exponent: \( 1 + \frac{r}{2} = 1.43577^{1/8} \). Calculate the eighth root: \( 1.43577^{1/8} \approx 1.045 \).

Step 5: Solve for Interest Rate

Subtract 1 from both sides to isolate \( \frac{r}{2} \): \( \frac{r}{2} = 1.045 - 1 \). Thus, \( \frac{r}{2} = 0.045 \). Multiply by 2 to solve for \( r \): \( r = 0.09 \).

Step 6: Convert Interest Rate to Percentage

Convert the decimal interest rate into a percentage by multiplying by 100: \( r = 0.09 \times 100 = 9\% \).

Key Concepts

Interest Rate Calculation

When it comes to understanding compound interest, one of the key elements is figuring out the interest rate. This piece of the puzzle determines how quickly your investment will grow.

Interest rate calculation involves understanding how much the original investment (principal) grows over a certain period due to this rate. It is usually expressed as a percentage, reflecting just how much more you'll have in the future compared to what you started with.

In this example, we have an investment that grows from $1000 to $1435.77 over four years via compounded semiannual interest. By knowing the starting amount, the ending amount, and the time frame, you can determine the interest rate. It tells you how efficiently your money is working for you each year, and it's a crucial concept for any investor.

Semiannual Compounding

Semiannual compounding means interest is calculated and added to the account balance twice a year. With each compounding period, the amount on which interest is earned becomes larger because the earned interest is added to the principal.

In our example, the interest is compounded semiannually. This means within those four years, the compounding happens a total of 8 times (2 times a year for 4 years). This adds more interest over the life of the investment compared to if it were compounded annually.

To understand this better, imagine if you only eat your favorite dessert once a year versus eating it twice. You'd have double the enjoyment! Similarly, your money grows faster with semiannual compounding because interest is added more frequently.

Compound Interest Formula

The compound interest formula is the mathematical magic that allows your money to grow exponentially over time. The formula is expressed as:\[A = P \left(1 + \frac{r}{n}\right)^{nt}\]where:

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

By rearranging this formula, you can solve for any unknown, like the interest rate. Plugging the values from the problem, you determined how this initial investment of \(1000 became \)1435.77 over four years with semiannual compounding. This formula captures the essence of compound interest, making it a powerful tool for predicting and maximizing future wealth.