Vinod makes a profit of Rs. 110 if he sells a certain number of pencils he has…

Concept:

When the same quantity of an item is sold at two different prices — one resulting in a profit and the other in a loss — against the same unknown cost price, subtracting the two profit/loss equations eliminates the cost price and isolates the quantity directly: (price₁ − price₂) × quantity = profit − (−loss) = profit + loss.

Application:

1. Let the number of pencils be x and the cost price per pencil be c.

2. Selling at Rs. 2.5 with a profit of 110 gives: (2.5 − c) × x = 110.

3. Selling at Rs. 1.75 with a loss of 55 gives: (1.75 − c) × x = −55.

4. Subtract the second equation from the first to eliminate c: (2.5 − 1.75) × x = 110 − (−55) ⇒ 0.75 × x = 165.

5. Solve for x: x = 165 ÷ 0.75 = 220.

Cross-check:

With x = 220, the cost price is c = 2.5 − 110/220 = Rs. 2.0 per pencil. In the loss scenario: revenue = 1.75 × 220 = Rs. 385, cost = 2.0 × 220 = Rs. 440, so the loss = 440 − 385 = Rs. 55, matching the given data. Both conditions are independently satisfied.

Therefore, Vinod has 220 pencils.