Question

A shop owner bought a total of 64 shirts from a wholesale market that came in two sizes, small and large. The price of a small shirt was INR 50 less than that of a large shirt.She paid a total of INR 5000 for the large shirts, and a total of INR 1800 for the small shirts.Then, the price of a large shirt and a small shirt together, in INR, is

• A. 150
• B. 225
• C. 175
• D. 200

Answer 1

The Correct Option is D

Let the number of small shirts be \(x\) and the price of each small shirt be \(y\).

Then, the number of large shirts is \(64 - x\) and the price of each large shirt becomes \(y + 50\).

Step 1: Form equations from the given conditions. 

Total money spent on small shirts: 
\(xy = 1800 \quad \cdots (1)\)

Total money spent on large shirts: 
\((64 - x)(y + 50) = 5000 \quad \cdots (2)\)

Step 2: Substitute equation (1) into equation (2).

Expand equation (2): 
\(64y + 3200 - xy - 50x = 5000\)

Using equation (1), \(xy = 1800\), substitute it: 
\(64y + 3200 - 1800 - 50x = 5000\)

Simplify: 
\(64y + 1400 - 50x = 5000\) 
\(64y - 50x = 3600\) 
Divide the entire equation by 2: 
\(32y - 25x = 1800 \quad \cdots (3)\)

Step 3: Substitute value of \( x \) from equation (1).

From (1): \(x = \frac{1800}{y}\) 
Substitute into (3): 
\(32y - 25 \cdot \left(\frac{1800}{y}\right) = 1800\)

Step 4: Clear the equation and simplify.

Multiply both sides by \(y\): 
\(32y^2 - 1800y - 25 \cdot 1800 = 0\) 
\(32y^2 - 1800y - 45000 = 0\)

Step 5: Solve the quadratic equation.

Divide the entire equation by 8: 
\(4y^2 - 225y - 5625 = 0\)

Solve this using the quadratic formula: 
\(y = \frac{225 \pm \sqrt{(-225)^2 + 4 \cdot 4 \cdot 5625}}{2 \cdot 4}\) 
\(y = \frac{225 \pm \sqrt{50625 + 90000}}{8} = \frac{225 \pm \sqrt{140625}}{8}\) 
\(y = \frac{225 \pm 375}{8}\) 
Only the positive value is valid: 
\(y = \frac{225 + 375}{8} = \frac{600}{8} = 75\)

Step 6: Final answer.

Price of a small shirt \(= y = 75\)
Price of a large shirt \(= y + 50 = 125\)
Total price of a small and a large shirt \(= 75 + 125 = 200\)

Correct Option: (D) 200