Simple interest split by alligation: ₹ 1000 is lent in two parts at 6% and 8% simple interest per annum. If the total annual interest is ₹ 75, how much was lent at 8%?

Introduction / Context:
With two rates and a known total interest, set up one linear equation in the unknown portion invested at the higher rate. This is a direct application of simple interest with a single time period (1 year).

Given Data / Assumptions:

• Total principal = ₹ 1000.
• Rates: 6% and 8% per annum.
• Total annual interest = ₹ 75.
• Let x be the amount at 8%.

Concept / Approach:
Interest = 0.08x + 0.06(1000 − x) = 75. Solve for x to get the principal at 8%.

Step-by-Step Solution:

Verification / Alternative check:
Check: 8% of 750 = 60; 6% of 250 = 15; total = 75, consistent.

Why Other Options Are Wrong:
The other values produce interest totals different from ₹ 75 upon substitution into 0.08x + 0.06(1000 − x).

Common Pitfalls:
Assuming equal split or averaging rates without regard to the interest total constraint.

Final Answer:
₹ 750