Jayant bought a certain number of white shirts at the rate of Rs per piece and a certain number of blue shirts at the rate of Rs per piece. For each shirt, he then set a fixed market price which was higher than the average cost of all the shirts. He sold all the shirts at a discount of and made a total profit of Rs . 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 per piece and a certain number of blue shirts at the rate of Rs per piece. For each shirt, he then set a fixed market price which was higher than the average cost of all the shirts. He sold all the shirts at a discount of and made a total profit of Rs . If he bought both colors of shirts, then the maximum possible total number of shirts that he could have bought is
Entered answer:
Solution
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 =
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 =
For x to be a positive integer, (3264 - 9y) must be divisible by 8.
Since 3264 = 8 × 408, we have:
x = 408 -
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