A sum of money doubles itself in 15 years at compound interest. In how many years it will become 8 times at the same compound interest rate ?

Question

Accepted Answer

To determine how many years it will take for a sum of money to become 8 times its original amount at the same compound interest rate, we start by understanding the given information: the money doubles itself in 15 years.

1. **Explanation for Option 4 (45 years):**

- We know that the money doubles in 15 years. This means that if the initial amount is $ P $, it becomes $ 2P $ in 15 years.
   - To find out how long it takes to become 8 times the original amount, we need to consider the doubling process:
     - After 15 years, the amount is $ 2P $.
     - After another 15 years (total 30 years), it doubles again to become $ 4P $.
     - After another 15 years (total 45 years), it doubles once more to become $ 8P $.
   - Therefore, it takes 45 years for the money to become 8 times its original amount. This supports Option 4 as the correct answer.

2. **Explanation for why other options are incorrect:**

- **Option 1 (30 years):**
     - In 30 years, the money doubles twice. Starting from $ P $, it becomes $ 2P $ in 15 years and $ 4P $ in 30 years. This is only 4 times the original amount, not 8 times. Therefore, Option 1 is incorrect.
   
   - **Option 2 (40 years):**
     - In 40 years, the money would not have completed another full doubling cycle beyond 30 years. It would be somewhere between $ 4P $ and $ 8P $, but not exactly $ 8P $. Therefore, Option 2 is incorrect.
   
   - **Option 3 (60 years):**
     - In 60 years, the money would have doubled four times: $ P $ to $ 2P $ (15 years), $ 2P $ to $ 4P $ (30 years), $ 4P $ to $ 8P $ (45 years), and $ 8P $ to $ 16P $ (60 years). This results in 16 times the original amount, which is more than 8 times. Therefore, Option 3 is incorrect.

In conclusion, the correct answer is Option 4 (45 years), as it accurately reflects the time required for the money to become 8 times its original amount at the given compound interest rate.