A sum of money invested at simple interest triples itself in 8 years. At the same rate of simple interest, how many times of the original principal will the amount become in 20 years time?

Introduction / Context:
This question uses a proportional reasoning approach with simple interest. Instead of directly giving the rate, the problem states that the amount becomes three times the original principal in 8 years. Under simple interest, the amount increase is linear in time, so once we know how many times the principal the amount becomes in one time frame, we can scale that relationship to a different time frame. Here we must ultimately find the multiple of the original principal after 20 years at the same simple interest rate.

Given Data / Assumptions:
Under simple interest, the amount becomes 3 times the principal in 8 years.
Rate of interest is constant and does not change over time.
We are asked for the factor by which the principal increases in 20 years at the same rate.
Interest type is simple interest, not compound interest.

Concept / Approach:
Let the principal be P. If the amount becomes 3P in 8 years, then the interest earned in 8 years is 3P minus P, which equals 2P. Under simple interest, I = P * r * t / 100. For t = 8, this becomes I = P * r * 8 / 100. Equating this to 2P allows us to find r, the annual rate. Once we know r, we can find the amount after 20 years using A = P * (1 + r * 20 / 100). However, we can also reason directly in terms of multiples once r is known.

Step-by-Step Solution:
Let the principal be P rupees.In 8 years the amount becomes 3P, so interest over 8 years is 3P - P = 2P.Simple interest over 8 years at rate r is I = P * r * 8 / 100.Set this equal to 2P: P * r * 8 / 100 = 2P.Cancel P on both sides to get r * 8 / 100 = 2.So r = 2 * 100 / 8 = 200 / 8 = 25 percent per annum.Now find the amount after 20 years at 25 percent simple interest.Amount after 20 years is A = P * (1 + r * 20 / 100).Compute r * 20 / 100 = 25 * 20 / 100 = 500 / 100 = 5.Thus A = P * (1 + 5) = P * 6.So in 20 years the amount becomes 6 times the original principal.

Verification / Alternative check:
We can confirm by checking the logic. If the rate is 25 percent per annum, then in 8 years the total interest is 8 * 25 percent = 200 percent of the principal, so the amount becomes 300 percent, or 3P, which matches the given condition. For 20 years, interest is 20 * 25 percent = 500 percent of the principal, so the amount becomes 600 percent, or 6P. This confirms that the factor in 20 years is 6 times the original principal.

Why Other Options Are Wrong:
8 times or 9 times would correspond to higher implied rates that would make the amount more than triple in 8 years, which contradicts the initial condition.
7 times is also too high and does not match the scaling based on a 25 percent annual rate.
5 times would correspond to a smaller total interest over 20 years than the 500 percent that follows from our calculated rate. Therefore, only 6 times is consistent with the given information.

Common Pitfalls:
Some learners mistakenly treat the tripling in 8 years as a compound interest scenario and use exponential reasoning, which is unnecessary here. Others miscalculate the rate by mixing up the steps while solving P * r * 8 / 100 = 2P. It is also easy to forget that the interest is in addition to the principal, so amount equal to 3P implies interest of 2P. Keeping these distinctions clear avoids mistakes.

Final Answer:
At the same simple interest rate, the amount becomes 6 times the original principal in 20 years.