On a particular day each of Danny, Edwin and Fahim sold three types of pens from their respective shops. Danny and Edwin sold an identical number of pens of Type A while Fahim sold twice as many pens of Type A as each of Danny and Edwin sold. The ratio of the numbers of pens of Type B sold by Danny, Edwin and Fahim was 3 : 5 : 2 respectively, while each of the trio sold an identical number of pens of Type C. The three sellers sold each of the types of pens at different prices per unit.

Assertion (A): It is possible that Danny sold each pen of Type A at a profit of Rs each pen of Type B at a loss of Re 1, and each pen of Type C at a loss of Rs 12 and made an overall profit of Rs 125; Edwin sold each pen of Type A at a profit of Rs 2, each pen of Type B at a loss of Rs 6, and each pen of Type C at a profit of Rs 6 and made an overall profit of Rs 110; and Fahim sold each pen of Type A at a profit of Re 1, each pen of Type B at a profit of Re 1, and each pen of Type C at a loss of Rs 3 and made an overall profit of Rs 210.

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

Correct option is C

Given:

Danny, Edwin, and Fahim sold three types of pens: Type A, Type B, and Type C.

Danny and Edwin sold an identical number of Type A pens, while Fahim sold twice as many Type A pens as Danny and Edwin.

The ratio of the number of Type B pens sold by Danny, Edwin, and Fahim was 3:5:2.

The number of Type C pens sold by each of them was the same.

Solution:

Let’s define the variables:

Let x be the number of Type A pens sold by Danny and Edwin (since Danny and Edwin sold the same number of Type A pens).

Let the number of Type B pens sold by Danny, Edwin, and Fahim be in the ratio 3:5:2, so we define:

3k pens for Danny,

5k pens for Edwin, and

2k pens for Fahim.

Let y be the number of Type C pens sold by each person (since all three sold the same number of Type C pens).

Danny's Profit Calculation:

Profit from Type A: 2x

Profit from Type B: -2 × 3k = -6k

Profit from Type C: -12y

Total profit for Danny:

2x - 6k - 12y = 125 ………..(1)

Edwin's Profit Calculation:

Profit from Type A: 2x

Profit from Type B: -6 × 5k = -30k

Profit from Type C: 6y

Total profit for Edwin:

2x - 30k + 6y = 110………..(2)

Fahim's Profit Calculation:

Profit from Type A: 4 × x = 4x

Profit from Type B: 8 × 2k = 16k

Profit from Type C: -3y

Total profit for Fahim:

4x + 16k - 3y = 210……………….(3)

Solving the system of equations:

From equation (1):

2x - 6k - 12y = 125

2x = 6k + 12y + 125

Substituting this expression for 2x into equations (2) and (3).

Substituting into equation (2):

(6k + 12y + 125) - 30k + 6y = 110

-24k + 18y = -15

-8k + 6y = -5 …………(4)

Substituting into equation (3):

4(6k + 12y + 125) + 16k - 3y = 210

24k + 48y + 500 + 16k - 3y = 210

40k + 45y = -290

8k + 9y = -58 ………………..(5)

Now, solving equations (4) and (5):

From equation (4):

-8k + 6y = -5

From equation (5):

88k + 9y = -58

Adding equations (4) and (5):

(6y + 9y) = -5 + (-58)

15y = -63

y = -\frac{63}{15} = -4.2

Since y represents the number of pens and must be a positive integer, this solution implies that the equations do not provide a valid solution for the given values.

Conclusion:

Assertion (A) is false, as the values provided do not yield a valid solution.

Reason (R) is true, as framing and solving linear equations is the correct approach, but in this case, the equations do not lead to a valid solution, indicating that Assertion (A) is not correct.

Thus, the correct answer is: Assertion (A) is false and Reason (R) is true.