Understanding Compound Interest Growth

Compound interest is a powerful concept where the interest earned is added back to the principal amount, and then the new total earns interest. This means your money grows at an accelerating rate over time.

The formula for compound interest is given by:

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

Where:

• A is the future value of the investment/loan, including interest
• P is the principal investment amount (the initial deposit or loan amount)
• r is the annual interest rate (as a decimal)
• n is the number of times that interest is compounded per year
• t is the number of years the money is invested or borrowed for

For simplicity, if the interest is compounded annually, the formula becomes:

$ A = P(1 + r)^t $

Calculating Time for Money Multiplication

Let's break down the problem step-by-step:

1. Initial Condition: Sum Triples

   We are told that a sum of money triples itself in 11 years. Let the initial principal amount be P. After 11 years, the amount A becomes 3P.

   Using the annual compound interest formula:

   $ 3P = P(1 + r)^{11} $

   Dividing both sides by P (assuming P is not zero), we get:

   $ 3 = (1 + r)^{11} $

   This equation gives us a relationship between the interest rate factor $(1+r)$ and the time.

2. Target Condition: Sum Becomes 27 Times

   We need to find the number of years (let's call it T) it will take for the sum to become 27 times the original principal amount (A = 27P).

   Using the same formula:

   $ 27P = P(1 + r)^T $

   Dividing both sides by P:

   $ 27 = (1 + r)^T $

3. Connecting the Conditions

   We know that 27 can be expressed as a power of 3:

   $ 27 = 3^3 $

   From the first condition, we found that $3 = (1 + r)^{11}$. We can substitute this into the equation for 27:

   $ 27 = \left[ (1 + r)^{11} \right]^3 $

   Using the rule of exponents $(a^m)^n = a^{m \times n}$, we get:

   $ 27 = (1 + r)^{11 \times 3} $

   $ 27 = (1 + r)^{33} $

4. Finding the Time (T)

   Now we have two equations for 27:

   • $ 27 = (1 + r)^T $
   • $ 27 = (1 + r)^{33} $

   By comparing these two equations, we can see that the exponents must be equal:

   $ T = 33 $

Therefore, it will take 33 years for the sum of money to amount to 27 times itself.