Simple interest on a principal sum for 3 years at 18% per annum is equal to half of the compound interest on ₹9,000 for 2 years at 10% per annum (compounded annually). What is the principal sum (in rupees) invested at simple interest?

Introduction:
This question combines simple interest (SI) and compound interest (CI) in a single equation. The main skill tested is computing CI for a given principal, rate, and time, then relating it to SI on an unknown principal. Because you must compute CI first, take half of it, and then solve for the SI principal using an SI expression, it involves multiple steps and is therefore hard.

Given Data / Assumptions:

• Unknown principal for SI = P
• SI rate = 18% per annum
• SI time = 3 years
• CI principal = ₹9,000
• CI rate = 10% per annum
• CI time = 2 years, compounded annually
• Given: SI(P) = (1/2) * CI(9000)

Concept / Approach:
Compute CI on ₹9,000 for 2 years at 10% compounded annually. CI = 9000 * [(1 + 10/100)^2 - 1]. Then take half of that value. Set SI on P, which is (P*18*3)/100, equal to that half-CI value and solve for P.

Step-by-Step Solution:
CI factor for 2 years at 10%: (1.10)^2 = 1.21 CI on 9000 = 9000 * (1.21 - 1) = 9000 * 0.21 = 1890 Half of CI = 1890 / 2 = 945 SI on P for 3 years at 18%: SI = (P * 18 * 3) / 100 = 0.54P Given 0.54P = 945 P = 945 / 0.54 = 1750

Verification / Alternative check:
If P = 1750, SI = (1750*18*3)/100 = 945. CI on 9000 is 1890, and half is 945, so the equality holds exactly.

Why Other Options Are Wrong:
Each incorrect principal would change SI proportionally (since SI is linear in P), so it would no longer equal 945, which is fixed by the CI part.

Common Pitfalls:
Computing amount instead of compound interest, using simple interest on ₹9000 instead of CI, or compounding incorrectly (not squaring 1.10).

Final Answer:
The principal invested at simple interest is ₹1,750.