Question

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

• A. 395
• B. 407
• C. 413
• D. None of Above

Answer 1

The Correct Option is B

Let's assume the following:

• Number of blue shirts: \( n \) 
• Number of white shirts: \( m \)
• Total number of shirts: \( m + n \)

Therefore, the total cost of the shirts is:

\(1000m + 1125n\)

The average cost of a shirt is:

\(\frac{1000m + 1125n}{m + n}\)

It is mentioned that the market price is set to be 25% higher than the average cost of all the shirts. Therefore, the selling price is:

\(\frac{1000m + 1125n}{m + n} \times \frac{5}{4} \times \frac{9}{10}\)

After simplifying, the average selling price of the shirts becomes:

\(\frac{9}{8} \times \frac{1000m + 1125n}{m + n}\)

The average profit per shirt is:

\(\frac{1}{8} \times \frac{1000m + 1125n}{m + n}\)

The total profit for all the shirts is:

\(\frac{1}{8} \times \frac{1000m + 1125n}{m + n} \times (m + n)\)

Which simplifies to:

\(\frac{1}{8} (1000m + 1125n)\)

The total profit is given as 51,000, so:

\(\frac{1}{8} (1000m + 1125n) = 51000\)

Multiplying both sides by 8:

\(1000m + 1125n = 408000\)

To maximize the total number of shirts, we need to minimize the value of \( n \), which can't be zero. Therefore, we need to maximize \( m \).

Rearranging the equation for \( m \):

\(m = \frac{408000 - 1125n}{1000}\)

Now, we maximize \( m \) by minimizing \( n \), and after inspection, we find that the maximum value for \( m \) is 399 and \( n \) is 8.

The total number of shirts is:

\(m + n = 399 + 8 = 407\)

Conclusion:

The maximum number of shirts is \( \boxed{407} \).

The correct option is (B): 407.