Compound Interest: Finding Time for Sixteen Times Amount

This problem involves calculating the time required for an investment to grow to a specific multiple of its initial value under compound interest.

Understanding the Compound Interest Principle

Compound interest means that interest earned is added to the principal, and future interest is calculated on this new, larger principal. The formula is:

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

Where:

• \( A \) = the future value of the investment/loan, including interest
• \( P \) = the principal investment amount (the initial deposit or loan amount)
• \( r \) = the annual interest rate (as a decimal)
• \( t \) = the time the money is invested or borrowed for, in years

Calculating Time for Investment Growth

We are given that the investment doubles in 6 years. Let the principal be \( P \). After 6 years, the amount \( A \) is \( 2P \).

Using the formula:

$ 2P = P(1 + r)^6 $

Dividing both sides by \( P \), we get the growth factor:

$ 2 = (1 + r)^6 $

We need to find the time \( T \) (in years) when the amount becomes sixteen times itself, i.e., \( A = 16P \).

$ 16P = P(1 + r)^T $

Dividing both sides by \( P \):

$ 16 = (1 + r)^T $

Determining the Final Time Period

We know that \( 16 \) can be expressed as a power of \( 2 \):

$ 16 = 2^4 $

Substitute the expression for \( 2 \) from the first condition ($ 2 = (1 + r)^6 $) into the equation for \( 16 \):

$ 16 = \left( (1 + r)^6 \right)^4 $

Using the rule of exponents $(a^m)^n = a^{mn}$, we simplify:

$ 16 = (1 + r)^{6 \times 4} $

$ 16 = (1 + r)^{24} $

Comparing this with the equation $ 16 = (1 + r)^T $, we can see that:

$ T = 24 \text{ years} $

Therefore, the amount will become sixteen times itself in 24 years.