A shopkeeper’s profit P is assumed to be a linear function of transportation charge t and quantity q: P = A*q + B*t (no constant term). He makes ₹10,000 by selling 20 units with t = ₹400, and ₹12,000 by selling 25 units with t = ₹600. Find the linear expression P(t, q).

Introduction / Context:
This is a modeling question in which profit is postulated to depend linearly on two variables: the quantity sold and the transportation charge. With two observed operating points, we can uniquely determine the coefficients of the linear model in the form P = A*q + B*t, assuming no constant term is needed (as directed by the options).

Given Data / Assumptions:

• P( q = 20, t = 400 ) = 10,000
• P( q = 25, t = 600 ) = 12,000
• Form: P = A*q + B*t

Concept / Approach:
Set up two linear equations in A and B using the two data points, then solve simultaneously. Because the model omits a constant term, the two unknowns map directly to the slopes with respect to q and t, respectively.

Step-by-Step Solution:
20A + 400B = 10,00025A + 600B = 12,000Subtract (first from second): (5A + 200B) = 2,000 ⇒ A + 40B = 400 ⇒ A = 400 − 40BPlug into 20A + 400B = 10,000 ⇒ 20(400 − 40B) + 400B = 10,0008,000 − 800B + 400B = 10,000 ⇒ −400B = 2,000 ⇒ B = −5Then A = 400 − 40(−5) = 400 + 200 = 600Hence P = 600q − 5t

Verification / Alternative check:
Check point 1: 600*20 − 5*400 = 12,000 − 2,000 = 10,000. Check point 2: 600*25 − 5*600 = 15,000 − 3,000 = 12,000. Both match exactly.

Why Other Options Are Wrong:
Other coefficient pairs do not satisfy both data points simultaneously; substituting will produce mismatches.

Common Pitfalls:
Including an unnecessary constant term; arithmetic slip when eliminating variables; forgetting that B can be negative (transport cost reduces profit).

Final Answer:
600q - 5t