Question

A sum of money becomes 4 times itself in 20 years at compound interest. In how many years will it become 16 times itself at the same rate?

• A. 30
• B. 35
• C. 40
• D. 45

Hint:

If money becomes \( n^2 \) times in a certain time, it will become \( n^4 \) times in double that time at the same compound interest rate.

Answer 1

The Correct Option is C

Step 1: Express the given information mathematically.
Let the principal be \( P \) and the rate of compound interest be \( r \).
According to the question:
\[ P(1+r)^{20} = 4P \] Dividing both sides by \( P \):
\[ (1+r)^{20} = 4 \]
Step 2: Relate 16 times to the given condition.
Since \( 4 = 2^2 \) and \( 16 = 2^4 \), we have:
\[ (1+r)^{20} = 2^2 \] To get \( 2^4 \), time required will be double.
Step 3: Calculate the required time.
\[ \text{Time} = 2 \times 20 = 40 \text{ years} \]
Step 4: Conclusion.
The sum becomes 16 times itself in 40 years at the same rate of compound interest.