In how many years will a sum become 27 times when it triples itself in 7 years at compound interest (when rates are same) ?

Question

Accepted Answer

To determine how many years it will take for a sum to become 27 times its original amount when it triples itself in 7 years at compound interest, we can use the properties of compound interest and the concept of exponential growth.

### Explanation for Option 4 (21 years):

1. **Understanding the Problem:**
   - The sum triples in 7 years. This means if the initial amount is $ P $, it becomes $ 3P $ in 7 years.
   - We need to find out how many years it will take for the sum to become $ 27P $.

2. **Compound Interest Growth:**
   - If a sum triples in 7 years, it means the compound interest rate allows the amount to grow by a factor of 3 every 7 years.
   - Therefore, in another 7 years (total 14 years), the amount will triple again, becoming $ 9P $ (since $ 3 \times 3 = 9 $).
   - In yet another 7 years (total 21 years), the amount will triple once more, becoming $ 27P $ (since $ 9 \times 3 = 27 $).

3. **Conclusion:**
   - Thus, it takes 21 years for the sum to become 27 times its original amount, confirming that Option 4 is correct.

### Explanation for Why Other Options are Incorrect:

- **Option 1 (18 years):**
  - In 18 years, the sum would have tripled twice (7 years + 7 years = 14 years), resulting in $ 9P $. It would not have reached $ 27P $ yet, as it requires another cycle of 7 years to triple again.

- **Option 2 (19 years):**
  - Similar to Option 1, in 19 years, the sum would still be in the second cycle of tripling, reaching only $ 9P $ by 14 years and not yet reaching $ 27P $.

- **Option 3 (20 years):**
  - In 20 years, the sum would still be in the process of the third cycle of tripling. It would not have completed the full 21 years needed to reach $ 27P $.

By following the logical progression of compound interest growth, we can see that only Option 4 (21 years) correctly results in the sum becoming 27 times its original amount.