Simple Interest – Two loans produce ₹ 90 total interest: A lends ₹ 600 to B for 2 years and ₹ 150 to C for 4 years, both at the same simple-interest rate. The total interest received is ₹ 90. Find the annual rate.

Introduction / Context:
When multiple loans share the same rate, their simple interests add linearly. The total interest allows solving for the common rate by summing each interest expression and equating to the given total.

Given Data / Assumptions:

• Loan 1: P1 = ₹ 600, t1 = 2 years
• Loan 2: P2 = ₹ 150, t2 = 4 years
• Total simple interest = ₹ 90
• Common rate r% per annum for both loans

Concept / Approach:
Total I = (P1 * r * t1 / 100) + (P2 * r * t2 / 100). Factor r and solve for it from the known total ₹ 90.

Step-by-Step Solution:
I_total = 600*r*2/100 + 150*r*4/100 = (1,200r + 600r)/100 = 1,800r/100 = 18rSet 18r = 90 ⇒ r = 90 / 18 = 5% per annum

Verification / Alternative check:
Compute interests at 5%: Loan 1 earns 600 * 5 * 2 / 100 = 60; Loan 2 earns 150 * 5 * 4 / 100 = 30; total = 90, as required.

Why Other Options Are Wrong:
4% yields only ₹ 72 total; 10% and 12% produce totals above ₹ 90; 6% gives ₹ 108, not ₹ 90.

Common Pitfalls:
Forgetting to apply the rate to each principal for its own time or omitting the /100 when converting percent to a fraction are typical mistakes.

Final Answer:
5%