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

\(\$2.1637 \times 10^{18}\)

Step-by-step solution

Identify the Given Information and Formula

To solve the problem, identify the principal amount (\(P\)), the interest rate (\(r\)), and the time period (\(t\)). For Methuselah, \(P = \$100\), \(r = 0.04\) (which is 4%), and \(t = 969\) years. The formula for compound interest when compounded annually is \(A = P(1 + r)^t\), where \(A\) is the amount after \(t\) years.

Substitute the Known Values

Using the compound interest formula \(A = P(1 + r)^t\), substitute the values: \(P = 100\), \(r = 0.04\), and \(t = 969\). The equation becomes:\[A = 100(1 + 0.04)^{969}\]

Calculate the Compound Interest

Calculate \((1 + 0.04)^{969}\), which equals approximately \(2.1637 \times 10^{16}\). Then multiply this by \(100\) to find \(A\). Thus:\[A \approx 100 \times (2.1637 \times 10^{16}) = 2.1637 \times 10^{18}\]

Determine the Final Amount

The final amount Methuselah would have is approximately \(\$2.1637 \times 10^{18}\).

Key concepts

Exponential Growth

Exponential growth is a powerful mathematical concept that describes situations where a quantity increases at a consistent rate over time. This type of growth is often seen in financial situations, such as with compound interest.

When we refer to exponential growth, we mean that the increase is proportionate to the current amount. Imagine a snowball rolling down a hill, gathering more snow as it goes – the bigger it gets, the more snow it collects. Compound interest works similarly, as the interest itself earns interest in subsequent periods.

In the context of Methuselah's savings, the original deposit grows exponentially because each year's interest calculations are based on not just the initial amount deposited, but also on any interest that has already been accrued. This results in a snowball effect, producing significant growth over long periods.

Exponential growth can be modeled using the formula:

• \( A = P(1 + r)^t \)

where:- \( A \) is the final amount,- \( P \) is the initial principal,- \( r \) is the interest rate per period,- \( t \) is the number of periods.

Understanding exponential growth helps in making strategic financial decisions, like saving for retirement or investing.

Financial Mathematics

Financial mathematics involves applying mathematical methods to solve problems related to finance. It's essential for analyzing and understanding how different financial instruments work. One key area where financial mathematics is used is in calculating interest rates and understanding the future value of investments.

At its core, financial mathematics allows individuals and businesses to make informed decisions about money management. By using formulas and models, it predicts how investments will grow over time. For example, by knowing the compound interest formula, you can determine how different factors – like changing the interest rate or the length of the investment – will affect the final amount.

In Methuselah's scenario, financial mathematics helps us calculate how his initial lifespan deposit of $100 grows into an astronomical sum over 969 years. It illustrates the profound impact time and interest rates have on the size of a future sum. This understanding of financial mathematics can be pivotal for anyone who wants to plan effectively for their financial future, such as saving for a child's education or planning for retirement.

Interest Rate Calculation

Interest rate calculation is a fundamental part of financial literacy, especially when it comes to understanding savings and borrowing costs. There are various types of interest rate calculations, but compound interest is particularly powerful because it takes into account the accumulation of interest over multiple periods.

Calculating compound interest involves determining how much interest is added to the principal each time, depending on the frequency of compounding. With annual compounding, as seen in Methuselah's example, the interest is calculated once per year and added to the principal.

To calculate the final amount with compound interest, the following formula is used:

• \( A = P(1 + r)^t \)

This formula shows how even a small change in the interest rate or time period can significantly impact the final amount. Each part of the formula serves a specific purpose:- \( P \) is the initial principal or deposit amount.- \( r \) is the annual interest rate, expressed as a decimal.- \( t \) is the total number of compounding periods, or years in this case.

By plugging in these values, you can easily predict how much a sum of money will grow over time. This skill is crucial for assessing investment opportunities and understanding the costs related to loans and mortgages.