A merchant purchases a cloth at a rate of Rs. 100 per meter and receives 5 cm length of cloth free for every 100 cm length of cloth purchased by him. He sells the same cloth at a rate of Rs. 110 per meter but cheats his customers by giving 95 cm length of cloth for every 100 cm length of cloth purchased by the customers. If the merchant provides a 5% discount, the resulting profit earned by him is
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Official Correct Answer: A. First, let me understand the problem. A merchant buys cloth at Rs. 100 per meter. For every 100 \text{ cm} he buys, he gets 5 \text{ cm} free. So, he's getting a little extra for free when he purchases. Then, he sells the cloth at Rs. 110 per meter, but he cheats his customers by giving them 95 \text{ cm} instead of 100 \text{ cm}. Also, he gives a 5% discount on the selling price. We need to find his resulting profit percentage.
Okay, let's structure this. I think We calculate the cost price per meter for the merchant, then the selling price per meter after the discount, and then find the profit percentage.
Starting with the cost price. The merchant buys at Rs. 100 per meter. But for every 100 \text{ cm} (which is 1 meter), he gets 5 \text{ cm} free. So, for every 100 \text{ cm}, he pays Rs. 100 but gets 105 \text{ cm} of cloth. That means his effective cost per meter is less because he's getting more cloth for the same price.
Calculate that. If he buys 100 \text{ cm}, he pays Rs. 100 and gets 105 \text{ cm}. So, the cost per \text{ cm} is 100 / 105. Therefore, the cost per meter (100 \text{ cm}) would be (100 / 105) * 100. Wait, no, that's not right. Wait, the cost per meter is actually the cost per 100 \text{ cm}. So, he pays Rs. 100 for 105 \text{ cm}. So, the cost per meter (100 \text{ cm}) is (100 / 105) * 100? Hmm, maybe I should think differently.
Wait, no. . For every 100 \text{ cm} he buys, he gets 5 \text{ cm} free. So, he's effectively paying for 100 \text{ cm} to get 105 \text{ cm}. So, the cost per \text{ cm} is 100 / 105. Therefore, the cost per meter (100 \text{ cm}) is 100 * (100 / 105). Wait, that doesn't make sense. Wait, no, the cost per meter is the cost for 100 \text{ cm}, which is Rs. 100, but he gets 5 \text{ cm} extra. So, the cost per meter is still Rs. 100, but he has more cloth. Hmm, maybe I'm overcomplicating.
Wait, perhaps I should calculate the cost price per meter for the merchant. Since he gets 5 \text{ cm} free for every 100 \text{ cm} purchased, his effective cost per meter is less. So, for 105 \text{ cm}, he pays Rs. 100. Therefore, the cost per meter is 100 / 1.05 meters. Wait, 100 Rs for 105 \text{ cm}, which is 1.05 meters. So, cost per meter is 100 / 1.05 ≈ 95.238 Rs per meter.
Okay, that makes sense. So, his cost price per meter is approximately Rs. 95.238.
Now, moving on to the selling price. He sells at Rs. 110 per meter, but he cheats by giving 95 \text{ cm} instead of 100 \text{ cm}. So, when a customer buys 100 \text{ cm}, the merchant gives them only 95 \text{ cm}. So, the merchant is selling 95 \text{ cm} for the price of 100 \text{ cm}. Therefore, the selling price per meter (100 \text{ cm}) is Rs. 110, but he's only giving 95 \text{ cm}. So, the effective selling price per meter is higher because he's giving less cloth.
Wait, no. Let me think again. The merchant sells at Rs. 110 per meter, but he gives 95 \text{ cm} for every 100 \text{ cm} the customer asks for. So, when a customer asks for 100 \text{ cm}, the merchant gives them 95 \text{ cm} and charges Rs. 110 for 100 \text{ cm}. So, the merchant is effectively selling 95 \text{ cm} for Rs. 110. Therefore, the selling price per meter (100 \text{ cm}) is Rs. 110, but he's only giving 95 \text{ cm}. So, the selling price per meter is actually higher in terms of cost.
Wait, maybe I should calculate the selling price per meter for the merchant. Since he sells 95 \text{ cm} for Rs. 110, the selling price per meter is (110 / 95) * 100. Because he's selling 95 \text{ cm} for Rs. 110, so per meter (100 \text{ cm}) it would be (110 / 95) * 100 ≈ 115.789 Rs per meter.
But wait, he also gives a 5% discount on the selling price. So, the selling price after discount is 95% of 115.789, which is approximately 110. So, wait, that can't be right. Wait, no, the discount is applied after the cheating. So, the merchant first cheats by giving 95 \text{ cm} for 100 \text{ cm}, and then gives a 5% discount on the selling price.
Wait, let me clarify. The selling price is Rs. 110 per meter, but he gives 95 \text{ cm} for every 100 \text{ cm}. So, the customer pays Rs. 110 for 95 \text{ cm}. Then, he gives a 5% discount on this Rs. 110. So, the selling price after discount is 110 * 0.95 = Rs. 104.5 per meter.
Wait, no, that doesn't make sense. Because the discount is on the selling price, which is already based on the 95 \text{ cm}. So, the customer pays Rs. 110 for 95 \text{ cm}, and then gets a 5% discount on that Rs. 110, making it Rs. 104.5 for 95 \text{ cm}. Therefore, the selling price per meter (100 \text{ cm}) would be (104.5 / 95) * 100 ≈ 110 Rs per meter. Wait, that seems circular.
I think I'm getting confused here. Let me try to structure this properly.
First, the merchant's cost