Explanation:

Let’s break it down step by step:

- We need the profit for each article, which is Selling Price – Cost Price.

- D is the easiest: For D, profit = Rs. 200, and SP = XY + 100, CP = Y²

So:

SP – CP = 200 ? XY + 100 – Y² = 200 ? XY – Y² = 100

- C gives a 20% discount:

Marked Price = (Y/2)³

Selling Price = Y² + XY – 5Y

Since discount is 20%, SP = 0.8 × MP,

Y² + XY – 5Y = 0.8 × (Y/2)³

You could solve for Y in terms of X, but the problem doesn’t demand it, since we want an expression for *profit*.

Profit (C): [Y² + XY – 5Y] – [XY] = Y² – 5Y

- B:

SP = X² + Y² + 75, CP = 2Y²

Profit = [X² + Y² + 75] – [2Y²] = X² – Y² + 75

- A:

SP = Y², CP = 4X + 5Y + 120

Profit = Y² – (4X + 5Y + 120) = Y² – 4X – 5Y – 120

Now, add all the profits:

- A: Y² – 4X – 5Y – 120

- B: X² – Y² + 75

- C: Y² – 5Y

- D: 200

Total profit = [Y² – 4X – 5Y – 120] + [X² – Y² + 75] + [Y² – 5Y] + 200

= X² + (Y² – Y² + Y²) – 4X – 5Y – 120 + 75 – 5Y + 200

= X² + Y² – 4X – 10Y + (–120 + 75 + 200)

= X² + Y² – 4X – 10Y + 155

Average = (Total profit) / 4

= (X² + Y² – 4X – 10Y + 155)/4

Now, let's look at all the options:

- Option 1: Rs. XY + 2X (no match)

- Option 2: Rs. 12X – 1.5Y (no match)

- Option 3: Rs. 4X + 5Y + 20 (this is not matching)

- Option 4: Rs. 35 × (X – Y) (no match)

- Option 5: None of these

Here’s the thing: *None* of these matches what we just got for the average.

So, the correct choice is Option 5: None of these.

Summary of statements/options:

- The problem wanted the average profit across four articles.

- We calculated this using the provided equations.

- The only option that fits our answer is "none of these."

- That’s it. You did the right process, but option 3 isn’t the right match.

Final Answer:

Option 5: None of these