In an aptitude question on simple interest with changing principal and rate, the simple interest on a certain principal for 16 years at a fixed annual rate is Rs. 10000. If instead the principal is doubled after every block of 4 years and, at the same times, the rate of interest is tripled while the initial principal and rate remain the same, what will be the total simple interest earned over the entire 16 year period?

Introduction / Context:
This aptitude question is based on the concept of simple interest with a twist: the principal and the rate of interest change at regular intervals of time. Instead of keeping the principal and rate constant for the whole 16 years, we are told that both are modified after every 4 year block. The question asks us to compare this new scenario with the original one, in which the simple interest earned in 16 years is Rs. 10000, and then compute the new total simple interest over the same overall period of 16 years.

Given Data / Assumptions:

• The simple interest on a certain principal for 16 years at a fixed annual rate is Rs. 10000.
• Let the principal be P and the annual rate be r percent.
• Time for the original situation = 16 years.
• In the new situation, the principal is doubled after every 4 years.
• At the same moments (every 4 years), the rate of interest is tripled.
• Interest throughout is simple interest, calculated separately for each 4 year block.

Concept / Approach:
The basic formula for simple interest is:
Simple interest = (Principal * Rate * Time) / 100.
From the original information we can find the product P * r, which will be useful. Then, for the modified case, we break the 16 years into four blocks of 4 years each. In every block, principal and rate are different, but time is the same (4 years). We compute the simple interest for each block in terms of P * r, add them up, and finally substitute the numerical value of P * r using the original simple interest information to obtain the new total interest.

Step-by-Step Solution:
Step 1: For the original situation, simple interest for 16 years is 10000, so using SI = (P * r * 16) / 100, we get (P * r * 16) / 100 = 10000. Step 2: Rearranging, P * r = (10000 * 100) / 16 = 1000000 / 16 = 62500. Step 3: Now consider the new situation. Divide the 16 years into four blocks of 4 years each: 0–4, 4–8, 8–12, and 12–16 years. Step 4: In years 0–4, principal = P and rate = r, so SI₁ = (P * r * 4) / 100 = (4/100) * (P * r). Step 5: After 4 years, principal doubles to 2P and rate triples to 3r. For years 4–8, SI₂ = (2P * 3r * 4) / 100 = (24/100) * P * r = (4/100) * 6 * P * r. Step 6: After 8 years, principal doubles again to 4P and rate triples again to 9r. For years 8–12, SI₃ = (4P * 9r * 4) / 100 = (144/100) * P * r = (4/100) * 36 * P * r. Step 7: After 12 years, principal becomes 8P and rate becomes 27r. For years 12–16, SI₄ = (8P * 27r * 4) / 100 = (864/100) * P * r = (4/100) * 216 * P * r. Step 8: Total new interest = SI₁ + SI₂ + SI₃ + SI₄ = (4/100) * P * r * (1 + 6 + 36 + 216) = (4/100) * P * r * 259. Step 9: Substitute P * r = 62500 to get total interest = (4/100) * 62500 * 259. Step 10: First compute (4/100) * 62500 = 2500. Then total interest = 2500 * 259 = 647500. Step 11: Therefore, the new simple interest over the entire 16 year period is Rs. 647500.

Verification / Alternative check:
We can quickly verify by comparing the factor of increase. Originally, over 16 years we had interest equal to (16/100) * P * r = 0.16 * P * r. In the new case, we effectively get (4/100) * P * r multiplied by 259, that is 0.04 * P * r * 259 = 10.36 * P * r. The ratio of new interest to original interest is 10.36 / 0.16 = 64.75, and 64.75 * 10000 = 647500, which matches the computed answer and confirms the calculation is consistent.

Why Other Options Are Wrong:
Rs. 632100 underestimates the effect of repeated doubling of principal and tripling of rate because it corresponds to using a smaller combined multiplier than 259. Rs. 689620 and Rs. 602260 come from incorrect intermediate sums or misapplied multipliers for each 4 year block. Rs. 615000 assumes a wrong factor for P * r or miscalculates one of the blocks. Only Rs. 647500 uses the correct sum of multipliers (1 + 6 + 36 + 216 = 259) and the correct value of P * r = 62500.

Common Pitfalls:
A common mistake is to treat this as a compound interest problem, but the question clearly specifies simple interest in each time segment. Another frequent error is to double the principal and triple the rate only once instead of after every 4 year block. Some students also forget to include all four blocks, or they fail to compute the factor sum correctly as 259. Finally, neglecting the initial step of finding P * r from the original simple interest relation can make the calculation unnecessarily complicated.

Final Answer:
The total simple interest earned when the principal is doubled and the rate is tripled after every 4 years over the full 16 year period is Rs. 647500.