A shopkeeper buys some pens. If he sells them at Rs 13 per pen, his total loss…

Concept: When the same quantity of an item is sold at two different prices — producing a loss at one price and a gain at the other — both situations share the same cost price and quantity. Writing each situation as an equation relating cost price, price per unit, and quantity, and then combining the two equations to eliminate the cost price, gives the quantity directly.

1. Let x = number of pens and y = total cost price (in Rs).

2. Selling at Rs 13 per pen gives revenue 13x. Since this is a loss of Rs 150, the cost price exceeds the revenue by Rs 150: y − 13x = 150 — equation (1).

3. Selling at Rs 15 per pen gives revenue 15x. Since this is a gain of Rs 100, the revenue exceeds the cost price by Rs 100: 15x − y = 100 — equation (2).

4. Adding equation (1) and equation (2) eliminates y: (y − 13x) + (15x − y) = 150 + 100, which simplifies to 2x = 250.

5. Dividing both sides by 2 gives x = 125.

Cross-check: Substituting x = 125 into equation (1) gives y = 13(125) + 150 = 1775. Checking this in equation (2): 15(125) − 1775 = 1875 − 1775 = 100, which matches the given gain — confirming the cost price and quantity are consistent. So the shopkeeper sold 125 pens.