Simple Interest – Equate two interest expressions to find the unknown rate: The simple interest on ₹ 4,000 for 3 years at R% per annum equals the simple interest on ₹ 5,000 at 12% per annum for 2 years. Find the value of R.

Introduction / Context:
When two simple-interest amounts are stated as equal, we can equate their formulas and solve for the unknown rate. This is a standard algebraic setup involving I = P * r * t / 100 for each case.

Given Data / Assumptions:

• Case 1: P1 = ₹ 4,000, time t1 = 3 years, rate = R% per annum
• Case 2: P2 = ₹ 5,000, time t2 = 2 years, rate = 12% per annum
• Simple interest: I = P * r * t / 100

Concept / Approach:
Set I1 = I2 and solve for R. This avoids computing the principal or amount; we only manipulate proportional quantities.

Step-by-Step Solution:
I1 = 4,000 * R * 3 / 100 = 120RI2 = 5,000 * 12 * 2 / 100 = 1,200Equate: 120R = 1,200 ⇒ R = 10

Verification / Alternative check:
Compute I1 with R = 10%: I1 = 4,000 * 10 * 3 / 100 = 1,200, which equals I2. Hence R = 10% is consistent.

Why Other Options Are Wrong:
6%, 8%, and 9% give less than ₹ 1,200 for Case 1; 12% gives more than required when substituted back, so they do not satisfy equality.

Common Pitfalls:
Using amounts instead of interests, or forgetting to divide by 100, leads to scale errors. Keep units and percent handling consistent.

Final Answer:
10%