Chained profits at different rates — Anupam → Bhargava → Chaudhary → Dara Singh. Anupam sells to Bhargava at a profit rate that is 4/5 of Bhargava’s profit rate to Chaudhary. Chaudhary sells to Dara at half the profit rate used by Anupam. If Chaudhary earns 10% profit when selling to Dara for ₹ 2805, what was Bhargava’s cost price for the painting (i.e., the price at which Bhargava bought it from Anupam)?

Introduction / Context:
This question chains three profit rates tied by ratios. We are given Chaudhary’s final sale to Dara (10% profit) and the exact selling price. Working backward gives Chaudhary’s cost (Bhargava’s selling price). Then, using the rate relationships, we recover Bhargava’s cost price (i.e., what he paid Anupam).

Given Data / Assumptions:

• Let Anupam’s profit rate = p1; Bhargava’s = p2; Chaudhary’s = p3.
• p1 = (4/5) * p2; p3 = (1/2) * p1.
• p3 = 10% and final selling price to Dara = ₹ 2805.

Concept / Approach:
First use p3 to find Chaudhary’s cost (which equals Bhargava’s selling price). Then determine p1 from p3, and p2 from p1. With p2 known, step back to Bhargava’s cost price (the price he paid to Anupam).

Step-by-Step Solution:

Verification / Alternative check:
Forward: 2040 * 1.25 = 2550 (Bhargava to Chaudhary); 2550 * 1.10 = 2805 (Chaudhary to Dara). All rates match the constraints.

Why Other Options Are Wrong:
1896, 1680, and 2000 do not maintain the chain rates when tested forward. 2160 corresponds to a different p2 value.

Common Pitfalls:
Mixing up “rate of profit” relationships versus absolute rupees, or forgetting that Chaudhary’s 10% is on his own cost (Bhargava’s selling price).

Final Answer:
2040