If a sum of money becomes double of itself in 8 years at simple interest, what is the rate of interest per annum, expressed as a percentage?

Introduction / Context:
This is a classic simple interest question about the time taken for a sum to double and the corresponding interest rate. Under simple interest, the amount increases linearly with time, so once we know that a sum doubles in a certain number of years, we can directly determine the annual rate of interest that produces this doubling.

Given Data / Assumptions:

• Principal (original sum) is P.
• Amount after 8 years is 2P, meaning the sum doubles in 8 years.
• Time T = 8 years.
• Rate of simple interest is R% per annum to be found.
• Simple interest formula: A = P * (1 + R * T / 100).

Concept / Approach:
Since A = 2P after 8 years, the total interest over those 8 years is P. Using the simple interest amount formula and equating it to 2P, we can solve for R. This uses the fact that, under simple interest, the extra amount over principal is proportional to time and rate: P * R * T / 100 = interest.

Step-by-Step Solution:
Given that amount after 8 years is A = 2P. Simple interest amount formula: A = P * (1 + R * T / 100). Substitute T = 8: 2P = P * (1 + 8R / 100). Divide both sides by P (P ≠ 0): 2 = 1 + 8R / 100. So 8R / 100 = 1, giving 8R = 100. Therefore, R = 100 / 8 = 12.5% per annum.
Verification / Alternative check:
If P = Rs. 100 for simplicity, then at 12.5% per annum simple interest, yearly interest is Rs. 12.50. Over 8 years, total interest = 8 * 12.50 = Rs. 100. The final amount after 8 years is 100 + 100 = Rs. 200, which is exactly double the original Rs. 100, confirming that 12.5% is correct.

Why Other Options Are Wrong:
Rates of 8.5%, 10%, 10.5%, and 14% produce different total interest over 8 years. For example, at 10%, total interest in 8 years is 80% of P, giving a final amount of 1.8P, which is less than double. At 14%, total interest is 112% of P, which overshoots the doubling condition.

Common Pitfalls:
Some learners confuse simple interest with compound interest and try to use doubling rules from compound interest. Others incorrectly divide 8 by 2 or use half of 8 in the formula. The correct method is to write out the simple interest amount expression, substitute A = 2P, and solve linearly for R.

Final Answer:
The rate of interest per annum is 12.5%.