Profit and Loss Question 4 - CAT 2023
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. 51,000. If he bought both colors of shirts, then the maximum possible total number of shirts that he could have bought is:
Step-by-Step Explanation
Let m be the number of white shirts and n be the number of blue shirts.
- Cost Metrics:
Total Cost Price = 1000m + 1125n
Average Cost Price per shirt = (1000m + 1125n) / (m + n)
- Revenue Metrics:
The market price is marked up by 25% (5 / 4), and a 10% discount (9 / 10) is applied to it.
Selling Price per shirt = Average CP × (5 / 4) × (9 / 10) = (9 / 8) × Average CP
Profit per shirt = SP − Average CP = 1 / 8 × Average CP
- Formulate the Total Profit Equation:
Total Profit = Profit per shirt × Total Shirts
51,000 = (1 / 8) × (1000m + 1125n)
408,000 = 1000m + 1125n
Simplify the equation by dividing both sides by 125:
8m + 9n = 3264
- Optimize the Variables (m,n > 0):
To maximize the total count m + n, we want to maximize m because it has the smaller coefficient (8), which allows for a larger total sum. Consequently, we must minimize n.
Rearranging the Diophantine equation:
8m = 3264 − 9n
Since 3264 and 8m are both perfectly divisible by 8, the term 9n must also be a multiple of 8. This implies n must be a multiple of 8.
The absolute minimum positive integer value for n is 8 (since Jayant purchased both colors, n ≠ 0).
Substitute n = 8:
8m = 3264 − 9 × 8
8m = 3192
m = 399
Maximum total shirts = m + n = 399 + 8 = 407
Correct Answer: 407
💡 FundaMakers Tip:
When handling Linear Diophantine Equations of the form ax + by = c:
- To maximize the sum x + y: Minimize the variable carrying the larger coefficient.
- Find the step structure: Use divisibility constraints. Since the remaining parts of the equation are divisible by 8, the term containing n must handle the balance by being a multiple of 8. This eliminates tedious guesswork and pinpoints the boundary condition instantly.