Question

If Methuselah's parents had put \(\$ 100\) in the bank for him at birth and he left it there, what would Methuselah have had at his death (969 years later) if interest was \(4 \%\) compounded annually?

Short answer

Methuselah would have about $1.69 \times 10^{17}$ dollars.

Step-by-step solution

Identify the Formula for Compound Interest

To solve the problem, use the compound interest formula: \[ A = P(1 + r)^n \]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, and \(n\) is the number of years the money is invested.

Assign Values to Variables

In the scenario described: - The principal \(P\) is \(\$100\).- The annual interest rate \(r\) is \(4\%\) which is equivalent to \(0.04\) as a decimal.- The time \(n\) is \(969\) years.

Substitute Values into Formula and Calculate

Substitute the values into the compound interest formula:\[ A = 100(1 + 0.04)^{969} \]Calculate the expression inside the parentheses first:\[ 1 + 0.04 = 1.04 \]Now calculate \(1.04^{969}\) and then multiply by \(100\) to find \(A\).

Calculate the Final Amount

Calculating \(1.04^{969}\) gives a very large number. After multiplying this by \(100\), you find that Methuselah would have approximately \(1.69 \times 10^{17}\) dollars.

Key concepts

Principal Amount

The principal amount is the initial investment or the starting amount of money you deposit in a financial account. In the context of our exercise, it is the sum of money that Methuselah's parents invested for him at birth. This is usually the first step when using the compound interest formula because it provides the base figure that will accrue interest over time. In our specific example, the principal amount is quite small—just $100. However, as you'll see, time and interest can significantly increase this initial sum. Understanding how to identify and manage the principal amount is crucial in financial planning because it lays the foundation for any future growth through interest. Consider this:

• The principal amount determines the starting point for interest calculations.
• It bears no interest until the first compounding period is completed, e.g., annually in our example.
• In long-term investments like Methuselah's, the initial principal might be small, but its growth can be exponential with time.

Annual Interest Rate

The annual interest rate is a critical factor in calculating compound interest. It represents the percentage of the principal that will be added to your account as interest every year. In mathematical terms, it's often represented as a decimal to simplify calculations in formulas. For instance, in our Methuselah example, the interest rate is given as 4%. This means every year, 4% of the current total amount is added to his account. Converting this to a decimal, you use 0.04 in the compound interest formula, making calculations more straightforward. Let’s break it down:

• The annual interest rate dictates the growth percentage each year.
• In calculations, always convert the percent to a decimal (e.g., 4% becomes 0.04).
• Higher interest rates increase the growth potential over long periods significantly.

Ultimately, the annual interest rate is a key predictor of how fast your investment will grow over time.

Exponential Growth

Exponential growth is a powerful concept in finance, particularly visible in compound interest calculations. Unlike simple interest, where the interest is applied only to the initial principal, compound interest adds interest on both the initial principal and the accumulated interest from previous periods. In Methuselah's case, exponential growth is what transforms a simple $100 into an astronomical sum over 969 years. The key to this growth is the repeated application of interest to an ever-increasing balance, which is described mathematically by raising a base (1 plus the annual interest rate) to a power corresponding to the number of periods (years, in this case). Understanding exponential growth involves these principles:

• Compound interest results in a constantly accelerating rate of growth, compared to linear growth.
• The formula uses exponents, hence the term 'exponential'.
• The longer the time frame, the more pronounced the exponential growth becomes.

This is why even relatively small investments can grow profoundly if left to accumulate over extensive periods, thanks to the power of exponential growth.