Let \(x\) represent the quantity of small shirts and \(y\) represent the price of each small shirt.
Consequently, the quantity of large shirts is \(64 - x\), and the price of each large shirt is \(y + 50\).
Step 1: Establish equations based on the provided information.
Expenditure on small shirts:
\(xy = 1800 \quad \cdots (1)\)
Expenditure on large shirts:
\((64 - x)(y + 50) = 5000 \quad \cdots (2)\)
Step 2: Integrate equation (1) into equation (2).
Expand equation (2):
\(64y + 3200 - xy - 50x = 5000\)
Substitute \(xy = 1800\) from equation (1):
\(64y + 3200 - 1800 - 50x = 5000\)
Simplify the equation:
\(64y + 1400 - 50x = 5000\)
\(64y - 50x = 3600\)
Divide the entire equation by 2:
\(32y - 25x = 1800 \quad \cdots (3)\)
Step 3: Isolate \(x\) from equation (1) and substitute it into equation (3).
From equation (1):
\(x = \frac{1800}{y}\)
Substitute this expression for \(x\) into equation (3):
\(32y - 25 \cdot \left(\frac{1800}{y}\right) = 1800\)
Step 4: Clear the denominator and simplify the equation.
Multiply both sides of the equation by \(y\):
\(32y^2 - 1800y - 25 \cdot 1800 = 0\)
\(32y^2 - 1800y - 45000 = 0\)
Step 5: Solve the resulting quadratic equation.
Divide the entire equation by 8:
\(4y^2 - 225y - 5625 = 0\)
Apply the quadratic formula to solve for \(y\):
\(y = \frac{-(-225) \pm \sqrt{(-225)^2 - 4(4)(-5625)}}{2(4)}\)
\(y = \frac{225 \pm \sqrt{50625 + 90000}}{8} = \frac{225 \pm \sqrt{140625}}{8}\)
\(y = \frac{225 \pm 375}{8}\)
Select the positive solution as price cannot be negative:
\(y = \frac{225 + 375}{8} = \frac{600}{8} = 75\)
Step 6: Determine the final answer.
Price of a small shirt:
\(= y = 75\)
Price of a large shirt:
\(= y + 50 = 75 + 50 = 125\)
Total price of one small and one large shirt:
\(= 75 + 125 = 200\)
Correct Option: (D) 200