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

Question

Accepted Answer

1. **Let the number of white shirts and blue shirts be $$ x $$ and $$ y $$, respectively.**  
   - Cost price of white shirts = $$ 1000x $$  
   - Cost price of blue shirts = $$ 1125y $$  
   - Total cost price = $$ 1000x + 1125y $$.

2. **Average cost price per shirt:**  
   $$
   	ext{Average cost per shirt} = \frac{1000x + 1125y}{x + y}.
   $$

3. **Market price per shirt (25% higher than average):**  
   $$
   	ext{Market price per shirt} = 1.25 	imes \frac{1000x + 1125y}{x + y}.
   $$

4. **Selling price per shirt (10% discount on market price):**  
   $$
   	ext{Selling price per shirt} = 0.9 	imes 1.25 	imes \frac{1000x + 1125y}{x + y} = 1.125 	imes \frac{1000x + 1125y}{x + y}.
   $$

5. **Total selling price for all shirts:**  
   $$
   	ext{Total selling price} = (x + y) 	imes 1.125 	imes \frac{1000x + 1125y}{x + y} = 1.125 	imes (1000x + 1125y).
   $$

6. **Profit earned by Jayant:**  
   Profit = Total selling price - Total cost price:  
   $$
   51000 = [1.125 	imes (1000x + 1125y)] - (1000x + 1125y).
   $$  
   Simplify:  
   $$
   51000 = [0.125 	imes (1000x + 1125y)].
   $$  
   $$
   51000 = 125x + 140.625y.
   $$

7. **Simplify further:**  
   Divide through by $$ 125 $$:  
   $$
   x + 1.125y = 408.
   $$

8. **Maximizing total number of shirts $$ x + y $$:**  
   Rewrite as:  
   $$
   x = 408 - 1.125y.
   $$  
   Since $$ x $$ and $$ y $$ are integers, $$ y $$ must be a multiple of $$ 8 $$ (to make $$ x $$ an integer). Test values:  
   
   - For $$ y = 8 $$:  
     $$
     x = 408 - (1.125 	imes 8) = 408 - 9 = 399.
     $$  
     Total number of shirts: $$ x + y = 399 + 8 = 407$$.

Thus, the maximum possible total number of shirts is **407**.