Problem 38 - Calculus: Graphical, Numerical, Algebraic

Question

A savings account earning compound interest triples in value in 10 years. How long will it take for the original investment to quadruple?

Step-by-Step Solution

Answer

The time it takes to quadruple the initial investment depends on the compound interest rate which is calculated based on the given information that the account triples in value in 10 years. Calculate this rate first and then apply it to when the account will quadruple.

Step 1: Determine the interest rate

Since we know that the original investment triples in 10 years, this means that \(A = 3P\), \(t = 10\), \(n = 1\) (assuming the interest is compounded annually) in the compound interest formula. Substitute these values into the formula and solve for \(r\), this will give the annual interest rate.

Step 2: Estimate the time for investment to quadruple

Now that the interest rate (\(r\)) is known, use it in the compound interest formula, this time by setting \(A = 4P\), to find out when the original investment quadruples. Substitute \(A = 4P\), \(n = 1\), and the previously calculated \(r\) into the formula and solve for \(t\).

Step 3: Calculate the result

Calculate the new \(t\) and the result provides the number of years it'll take for the original investment to quadruple under the same annual interest rate.

Key Concepts

Tripling and Quadrupling Investments

Investing money with compound interest can significantly increase its value over time. One fascinating aspect is how an investment can grow to double, triple, or even quadruple its original amount. Here, the focus is on tripling and quadrupling.
Suppose an investment triples in 10 years with a specific interest rate. This tells us how much the principal, or original amount, grows. If it can triple, it might also quadruple with this same rate.
The idea is to understand how long this additional growth takes. By using compound interest formulas, which we'll explore next, we can determine the time for both tripling and quadrupling. This understanding fundamentally relies on knowing the exact interest rate and how compound interest works over time.

Interest Rate Calculation

Calculating the interest rate is crucial for understanding how investments can grow. Compound interest is computed using the formula:
\[A = P \left(1 + \frac{r}{n}\right)^{nt}\]where:

• \(A\) is the future value of the investment,
• \(P\) is the principal investment amount,
• \(r\) is the annual interest rate (decimal),
• \(n\) is the number of times that interest is compounded per year, and
• \(t\) is the time the money is invested for in years.

Given that an investment triples in 10 years, we set \(A = 3P\) and \(n = 1\), which assumes the interest is compounded annually and solve for \(r\).
This step involves some algebra to rearrange the formula and accurately calculate \(r\), giving us the insight needed to project future growth, such as quadrupling the investment.

Time Estimation for Investments

Estimating the time it takes for an investment to reach a certain value is a key financial skill. After determining the interest rate, we apply it to find out how quickly the investment grows.
To find the time it takes to quadruple the original investment, we set \(A = 4P\) in the compound interest formula:
\[4P = P \left(1 + \frac{r}{n}\right)^{nt}\]By solving for \(t\) using the previously calculated \(r\), we isolate time to see how long it takes for our investment to grow fourfold.
This involves logarithmic calculations but ultimately gives us the timeframe needed based on the same annual interest rate. Understanding this can help in planning future investments and setting realistic financial goals.