Compound Interest Sum Growth Calculation

This problem concerns the exponential growth of a sum under compound interest.

Understanding Growth Factors

Let the initial principal sum be denoted by P.

• The problem states that the sum becomes 4 times its initial value (4P) in a period of 3 years.
• This implies a growth factor of 4 over every 3-year period, assuming a constant interest rate.
• The goal is to determine the time required for the sum to become 64 times its initial value (64P).

Calculating Time for 64 Times Growth

We need to relate the target growth factor (64) to the known growth factor (4).

• We can express 64 as a power of 4: $ 64 = 4 \times 4 \times 4 = 4^3 $
• This means the process that multiplies the sum by 4 needs to be repeated 3 times to achieve a total multiplication factor of 64.
• Since each 3-year period results in the sum multiplying by 4, three such periods are required.
• The total time is calculated as: $ \text{Total Time} = (\text{Number of 4x growth cycles}) \times (\text{Time per cycle}) $ $ \text{Total Time} = 3 \times 3 \text{ years} $ $ \text{Total Time} = 9 \text{ years} $

Thus, the sum will become 64 times its original value in 9 years.