Average rate planning across four parts — find the missing rate: ₹ 10000 is lent in four parts: ₹ 2000 at 8%, ₹ 4000 at 7.5%, ₹ 1400 at 8.5%, and the remainder at an unknown rate r. If the overall average rate is 8.13% p.a., what is r?

Introduction / Context:
Weighted averages convert a mix of different rates on different principal slices into a single effective rate. One unknown slice can be found by equating total interest to the target average interest.

Given Data / Assumptions:

• Total principal = ₹ 10000
• Parts: ₹ 2000 @ 8%, ₹ 4000 @ 7.5%, ₹ 1400 @ 8.5%, remainder @ r
• Overall average rate = 8.13% → annual interest = ₹ 813
• Remainder amount = 10000 - (2000 + 4000 + 1400) = ₹ 2600

Concept / Approach:
Total annual interest equals the sum of each part’s annual interest. Solve for r so that the sum equals ₹ 813.

Step-by-Step Solution:
Known interest: 2000 * 0.08 = 160; 4000 * 0.075 = 300; 1400 * 0.085 = 119Subtotal = 160 + 300 + 119 = ₹ 579Needed from remainder = 813 - 579 = ₹ 234Let r be the percent rate for ₹ 2600: 2600 * r / 100 = 234 → r = 9%

Verification / Alternative check:
Total interest: 579 + 234 = 813 → 813 / 10000 = 8.13% p.a. (as required).

Why Other Options Are Wrong:
Values below or above 9% make the total interest deviate from ₹ 813, hence the average would not be 8.13%.

Common Pitfalls:
Arithmetic mistakes in partial interests or forgetting that average rate × total principal = total interest.

Final Answer:
9%