CI vs SI on Same Sum and Rate — Recover principal from 2-year totals: The simple interest on a certain sum for 2 years is ₹ 40, while the compound interest on the same sum for the same rate and 2 years is ₹ 45. Find the principal.

Introduction / Context:
For two years at the same annual rate r (decimal), SI(2y) = 2 P r, and CI(2y) = 2 P r + P r^2. The difference isolates the interest-on-interest term P r^2, letting us solve for r and/or P using the two totals provided.

Given Data / Assumptions:

• SI over 2 years = ₹ 40 ⇒ 2 P r = 40
• CI over 2 years = ₹ 45 ⇒ 2 P r + P r^2 = 45
• Same sum and same rate for both measures

Concept / Approach:
Subtract to get P r^2 = 5. From 2 P r = 40 infer P r = 20. Then r = (P r^2)/(P r) = 5 / 20 = 0.25 (25%). Finally, compute P = (P r)/r = 20 / 0.25 = 80.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• ₹ 70, ₹ 85, ₹ 90, ₹ 75 do not simultaneously satisfy SI = 40 and CI = 45 at one common rate.

Common Pitfalls:

• Mistaking the 5 difference as the rate itself; it represents P r^2, not r.

Final Answer:
Rs. 80.