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Jayant bought a certain number of white shirts at the rate of Rs 10001000 per piece and a certain number of blue shirts at the rate of Rs 11251125 per piece. For each shirt, he then set a fixed market price which was 25%25 \% higher than the average cost of all the shirts. He sold all the shirts at a discount of 10%10 \% and made a total profit of Rs 5100051000. If he bought both colors of shirts, then the maximum possible total number of shirts that he could have bought is

Entered answer:

Solution

✅ Correct Answer: 407

Let us define variables clearly:

Number of white shirts = x

Number of blue shirts = y

We need to find the maximum value of (x + y)


Total Cost Price (CP):

CP = 1000x + 1125y

Average Cost Per Shirt:

Average cost = 1000x+1125yx+y\tfrac{1000x + 1125y}{x + y}

Market Price (MP) - 25% higher than average cost:

Since MP is 25% higher than average cost:

MP per shirt = 1.25 × Average cost

Total MP = 1.25 × (1000x + 1125y)

Selling Price (SP) - 10% discount on MP:

SP per shirt = 90% of MP = 0.9 × MP per shirt

Total SP = 0.9 × 1.25 × (1000x + 1125y) = 1.125(1000x + 1125y)


Key Insight: Profit = Selling Price - Cost Price

Total Profit = 1.125(1000x + 1125y) - (1000x + 1125y)

51000 = 0.125(1000x + 1125y)

1000x + 1125y = 51000/0.125

= 408000

Dividing by 125:

8x + 9y = 3264


From 8x + 9y = 3264, we can express x in terms of y:

8x = 3264 - 9y

x = 3264−9y8\tfrac{3264 - 9y}{8}

For x to be a positive integer, (3264 - 9y) must be divisible by 8.

Since 3264 = 8 × 408, we have:

x = 408 - 9y8\tfrac{9y}{8}

For x to be an integer, y must be divisible by 8.


Let y = 8k where k is a positive integer.

Then: x = 408 - 9k

Constraints:

x > 0 ⟹ 408 - 9k > 0 ⟹ k < 408/9 ≈ 45.33

y > 0 ⟹ k > 0

So k can be 1, 2, 3, ..., 45.

Total shirts = x + y = (408 - 9k) + 8k = 408 - k

To maximize total shirts, we need to minimize k.

The minimum value of k = 1.

Therefore:

y = 8(1) = 8

x = 408 - 9(1) = 399

Maximum total shirts = 399 + 8 = 407


Answer: 407

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