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
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Official Correct Answer: 407. Okay, so I'm trying to solve this CAT question about Jayant buying shirts and making a profit. Let me read it again and break it down step by step.
Jayant bought white shirts at Rs 1000 each and blue shirts at Rs 1125 each. He set a market price that was 25% higher than the average cost of all shirts. Then he sold them at a 10% discount and made a total profit of Rs 51000. We need to find the maximum possible total number of shirts he could have bought.
First, let's note down the given information:
- Cost price of white shirts (Cw) = Rs 1000 per piece
- Cost price of blue shirts (Cb) = Rs 1125 per piece
- Market price (Mp) = 25% higher than the average cost price
- Selling price (Sp) = Market price - 10% discount on Mp
- Total profit = Rs 51000
- He bought both colors, so the number of white and blue shirts is at least 1 each.
We need to find the maximum total number of shirts, which is white shirts (w) + blue shirts (b).
Let me define variables:
Let w = number of white shirts
Let b = number of blue shirts
Total cost (C) = 1000w + 1125b
Average cost per shirt (A) = Total cost / Total shirts = (1000w + 1125b) / (w + b)
Market price (Mp) = A + 25% of A = 1.25A
Selling price (Sp) = Mp - 10% of Mp = 0.9 * Mp = 0.9 * 1.25A = 1.125A
Total selling price (S) = Sp * total shirts = 1.125A * (w + b)
Profit = S - C = 51000
So, let's write the equation:
1.125A*(w + b) - (1000w + 1125b) = 51000
But A is (1000w + 1125b)/(w + b), so let's substitute that:
1.125 * [(1000w + 1125b)/(w + b)] * (w + b) - (1000w + 1125b) = 51000
Simplify the first term:
1.125*(1000w + 1125b) - (1000w + 1125b) = 51000
Factor out (1000w + 1125b):
(1.125 - 1)*(1000w + 1125b) = 51000
0.125*(1000w + 1125b) = 51000
Multiply both sides by 8:
1000w + 1125b = 51000 * 8 = 408000
So, 1000w + 1125b = 408000
We can divide the entire equation by 25 to simplify:
40w + 45b = 16320
Let me write this as:
40w + 45b = 16320
We can simplify further by dividing by 5:
8w + 9b = 3264
So, 8w + 9b = 3264
We need to find integer solutions for w and b where w and b are positive integers.
We can express this as:
8w = 3264 - 9b
So, w = (3264 - 9b)/8
Since w must be an integer, (3264 - 9b) must be divisible by 8.
Let's find the condition for divisibility by 8.
3264 divided by 8 is 408, which is an integer.
So, 9b must also leave a remainder such that (3264 - 9b) is divisible by 8.
Compute 9b mod 8.
9 mod 8 = 1, so 9b mod 8 = b mod 8.
So, (3264 - 9b) mod 8 = (0 - b mod 8) mod 8 = (-b mod 8) = (8 - b mod 8) if b mod 8 ≠ 0.
For (3264 - 9b) to be divisible by 8, (3264 - 9b) mod 8 = 0.
Which implies that b mod 8 = 0.
So, b must be a multiple of 8.
Let me set b = 8k, where k is a positive integer.
Then, substituting back into the equation:
8w + 9*(8k) = 3264
8w + 72k = 3264
Divide both sides by 8:
w + 9k = 408
So, w = 408 - 9k
Since w must be positive, 408 - 9k > 0
So, 9k < 408
k < 408/9 ≈ 45.333
So, k can be from 1 to 45.
But we need to maximize the total number of shirts, which is w + b = (408 - 9k) + 8k = 408 - k
To maximize 408 - k, we need to minimize k.
The smallest k is 1.
So, when k = 1:
w = 408 - 9*1 = 399
b = 8*1 = 8
Total shirts = 399 + 8 = 407
Wait, but let me check if k can be 0. If k=0, then b=0, but the question says he bought both colors, so b must be at least 1. So k starts at 1.
Is 407 the maximum possible? Let me check for k=1, total shirts=407.
If k=2, total shirts=406, which is less.
So, yes, the maximum total number of shirts is 407.
I think that's the answer.
Total cost: 1000*399