Jayant bought a certain number of white shirts at the rate of Rs 1000 per piece and a certain number of blue shirts at the rate of Rs 1125 per piece. For each shirt, he then set a fixed market price which was 25% higher than the average cost of all the shirts. He sold all the shirts at a discount of 10% and made a total profit of Rs 51000. If he bought both colors of shirts, then the maximum possible total number of shirts that he could have bought is:
Question:
Jayant bought a certain number of white shirts at the rate of Rs 1000 per piece and a certain number of blue shirts at the rate of Rs 1125 per piece. For each shirt, he then set a fixed market price which was 25% higher than the average cost of all the shirts. He sold all the shirts at a discount of 10% and made a total profit of Rs 51000. If he bought both colors of shirts, then the maximum possible total number of shirts that he could have bought is:
Jayant bought a certain number of white shirts at the rate of Rs 1000 per piece and a certain number of blue shirts at the rate of Rs 1125 per piece. For each shirt, he then set a fixed market price which was 25% higher than the average cost of all the shirts. He sold all the shirts at a discount of 10% and made a total profit of Rs 51000. If he bought both colors of shirts, then the maximum possible total number of shirts that he could have bought is:
Options
Answer: 407
Explanation:
Step 1: Define variables Let x = number of white shirts at Rs 1000 each Let y = number of blue shirts at Rs 1125 each Total cost = 1000x + 1125y Step 2: Market price and selling price Average cost per shirt = (1000x + 1125y) / (x + y) Marked price = 25% higher than average cost = 1.25 * (1000x + 1125y) / (x + y) Selling price after 10% discount = 0.9 * Marked price = 1.125 * (1000x + 1125y) / (x + y) Step 3: Total profit Total SP = (x + y) * 1.125 * (1000x + 1125y) / (x + y) = 1.125 * (1000x + 1125y) Profit = SP - Cost = 0.125 * (1000x + 1125y) Given profit = 51000, so: 0.125 * (1000x + 1125y) = 51000 1000x + 1125y = 408000 Step 4: Simplify equation Divide by 25: 40x + 45y = 16320 Divide by 5: 8x + 9y = 3264 Step 5: Find integer solutions Solve for x in terms of y: x = (3264 - 9*y) / 8 x must be integer ⇒ 3264 - 9*y divisible by 8 ⇒ y divisible by 8 Let y = 8k, then x = 408 - 9k Both x > 0 and y > 0 ⇒ k ≥ 1 and k ≤ 45 Step 6: Maximize total number of shirts Total shirts = x + y = (408 - 9k) + (8k) = 408 - k To maximize total shirts, minimize k = 1 Then: y = 81 = 8 x = 408 - 91 = 399 Total shirts = 399 + 8 = 407 Answer: 407
Explanation:
Step 1: Define variables Let x = number of white shirts at Rs 1000 each Let y = number of blue shirts at Rs 1125 each Total cost = 1000x + 1125y Step 2: Market price and selling price Average cost per shirt = (1000x + 1125y) / (x + y) Marked price = 25% higher than average cost = 1.25 * (1000x + 1125y) / (x + y) Selling price after 10% discount = 0.9 * Marked price = 1.125 * (1000x + 1125y) / (x + y) Step 3: Total profit Total SP = (x + y) * 1.125 * (1000x + 1125y) / (x + y) = 1.125 * (1000x + 1125y) Profit = SP - Cost = 0.125 * (1000x + 1125y) Given profit = 51000, so: 0.125 * (1000x + 1125y) = 51000 1000x + 1125y = 408000 Step 4: Simplify equation Divide by 25: 40x + 45y = 16320 Divide by 5: 8x + 9y = 3264 Step 5: Find integer solutions Solve for x in terms of y: x = (3264 - 9*y) / 8 x must be integer ⇒ 3264 - 9*y divisible by 8 ⇒ y divisible by 8 Let y = 8k, then x = 408 - 9k Both x > 0 and y > 0 ⇒ k ≥ 1 and k ≤ 45 Step 6: Maximize total number of shirts Total shirts = x + y = (408 - 9k) + (8k) = 408 - k To maximize total shirts, minimize k = 1 Then: y = 81 = 8 x = 408 - 91 = 399 Total shirts = 399 + 8 = 407 Answer: 407
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