On a particular day each of Danish, Ethan and Farhan sold three types of pens from their respective shops. Danish and Ethan sold an identical number of pens of Type A while Farhan sold $$\frac{3}{2}$$​ times as many pens of Type A as Danish and Ethan together sold. The ratio of the numbers of pens of Type B sold by Danish, Ethan and Farhan was 4 : 3 : 1 respectively, and Danish and Farhan sold an identical number of pens of Type C each, while Ethan sold $$\frac{5}{2}$$​ as many pens of Type C as each of Ethan and Farhan sold. The three sellers sold each of the types of pens at different prices per unit.

Assertion (A): It is possible that Danish sold each pen of Type A at a loss of Re 1, each pen of Type B at a profit of Rs 5, and each pen of Type C at a loss of Rs 3 and made an overall profit of Rs 114; Ethan sold each pen of Type A at a profit of Rs 4, each pen of Type B at a loss of Rs 3, and each pen of Type C at a profit of Rs 5 and made an overall profit of Rs 267; and Farhan sold each pen of Type A at a profit of Rs 8, each pen of Type B at a profit of Rs 5, and each pen of Type C at a loss of Rs 7 and made an overall profit of Rs 60.

Reason (R): Framing and solving the three possible linear equations we will find that we get a unique solution.

Correct option is B

Given:
Danish and Ethan each sold x pens of Type A.
Farhan sold 3 times x pens of Type A.
Type B pens sold in the ratio 4 : 3 : 1 by Danish, Ethan, and Farhan. So numbers are 4y, 3y, and y respectively.
Type C pens:
Danish = z
Farhan = z
Ethan = 5/2 × z

Formula:

$$\text{Successive Replacement Formula:} \\\text{Final Quantity} = \text{Initial Quantity} \times \left(1 - \frac{y}{V}\right)^n\\\text{Profit/Loss Formula:} \\\text{Total Profit or Loss} = \sum (\text{Number of items} \times \text{Profit or Loss per item})$$$$\text{Successive Replacement Formula:} \\\text{Final Quantity} = \text{Initial Quantity} \times \left(1 - \frac{y}{V}\right)^n\\\text{Profit/Loss Formula:} \\\text{Total Profit or Loss} = \sum (\text{Number of items} \times \text{Profit or Loss per item})$$​

Solution:

​Form Profit Equations:
Danish:

Sells each Type A pen at a loss of ₹1 → Total = -1 × x
Sells each Type B pen at a profit of ₹5 → Total = 5 × 4y = 20y
Sells each Type C pen at a loss of ₹3 → Total = -3 × z
Overall Profit = ₹114

Equation:

x + 20y - 3z = 114

Ethan:

Sells each Type A pen at a profit of ₹4 → 4 × x
Sells each Type B pen at a loss of ₹3 → -3 × 3y = -9y
Sells each Type C pen at a profit of ₹5 → 5 × (5/2) × z = 25/2 × z

Overall Profit = ₹267

Equation: 4x - 9y + (25/2)z = 267
Multiply the whole equation by 2 to avoid fractions: 8x - 18y + 25z = 534

Farhan:

Sells each Type A pen at a profit of ₹8 → 8 × 3x = 24x
Sells each Type B pen at a profit of ₹5 → 5 × y = 5y
Sells each Type C pen at a loss of ₹7 → -7 × z
Overall Profit = ₹60

Equation:
24x + 5y - 7z = 60

System of Three Equations Formed:
-x + 20y - 3z = 114

8x - 18y + 25z = 534

24x + 5y - 7z = 60

Conclusion:
We have 3 independent equations in 3 unknowns (x, y, z).
The equations are not multiples of each other.
Therefore, they will give a unique solution.
Hence, Reason (R) is true.
Also, as the structure of the profit situation is correctly set up and consistent with the profits given, Assertion (A) is true.
However, Reason (R) (about unique solvability) is not the direct explanation for Assertion (A) (about specific profits).

Final Answer:
Both Assertion (A) and Reason (R) are true, but Reason (R) is not a correct explanation of Assertion (A).
Thus, the answer is:
(B)