A certain sum of money amounts to Rs. 1,008 in 2 years and Rs. 1,164 in 3.5 years at simple interest. Find the principal sum and the rate of interest per annum.

Introduction / Context:
This problem combines two different time period amounts under simple interest to determine both the principal and the interest rate. It is a useful pattern because it demonstrates how constant yearly interest under simple interest can be extracted from the difference between amounts at two different times. Once the annual interest is known, both the principal and the rate can be found easily.

Given Data / Assumptions:

Amount after 2 years A2 = Rs. 1,008.
Amount after 3.5 years A3.5 = Rs. 1,164.
Interest is simple interest, not compound.
We must find the principal P and the annual rate of interest r percent.

Concept / Approach:
Under simple interest, interest is proportional to time, with the same amount of interest added each year. Therefore, the difference between the amounts at two different times gives the interest for the additional time between those points. Here, the time difference is 1.5 years. Dividing the extra interest by 1.5 gives the interest per year, from which we can get the annual rate and the principal.

Step-by-Step Solution:
Let yearly simple interest be I rupees per year. Amount after 2 years: A2 = P + 2I = 1,008. Amount after 3.5 years: A3.5 = P + 3.5I = 1,164. Difference in amount over the additional 1.5 years is A3.5 - A2 = 1,164 - 1,008 = 156. This difference of 156 is the interest for 1.5 years, so 1.5I = 156. Therefore I = 156 / 1.5 = 104 rupees per year. Now substitute back into A2 = P + 2I. So 1,008 = P + 2 * 104 = P + 208. Thus P = 1,008 - 208 = 800 rupees. To find the rate, use I = (P * r) / 100 for 1 year, so 104 = (800 * r) / 100. Hence r = (104 * 100) / 800 = 13 percent per annum.

Verification / Alternative check:
Check using P = 800 and r = 13 percent. Yearly interest = (800 * 13) / 100 = 104. After 2 years, interest = 2 * 104 = 208, so amount = 800 + 208 = 1,008, which matches A2. After 3.5 years, interest = 3.5 * 104 = 364, so amount = 800 + 364 = 1,164, matching A3.5. This confirms that P = 800 and r = 13 percent are correct.

Why Other Options Are Wrong:
Option 800, 14 percent would produce yearly interest of 112, leading to amounts that do not match 1,008 and 1,164 at 2 and 3.5 years respectively.
Option 800, 12 percent would give yearly interest of 96, again resulting in mismatched amounts at the stated times.
Option 800, 19 percent would yield even larger yearly interest and clearly overshoot the given amounts.

Common Pitfalls:
Some learners mistakenly average the two times or two amounts without using the difference to find yearly interest.
Others miscalculate 1.5I = 156 and incorrectly divide by 2 instead of 1.5, leading to a wrong yearly interest.
Another error is to confuse the simple interest formula with compound interest and attempt to apply exponential growth, which is not appropriate here.

Final Answer:
The principal is Rs. 800 and the rate of interest is 13 percent per annum.