Step-by-Step Solution

Step-by-Step Solution

Step 1: Define variables and formulate rate equation

Let $x$ be the number of dozen eggs sold ($x > 0$).

• Original rate per dozen $= \dfrac{2400}{x}$
• Hypothetical rate per dozen $= \dfrac{2400}{x + 2}$

Given that the price is Rs. $0.50 = \dfrac{1}{2}$ cheaper per dozen: $\frac{2400}{x} - \frac{2400}{x + 2} = \frac{1}{2}$

Step 2: Simplify and form the quadratic equation

Combine fractions on the left: $\frac{2400(x + 2) - 2400x}{x(x + 2)} = \frac{1}{2}$

$\frac{4800}{x^2 + 2x} = \frac{1}{2}$

Cross-multiply: $x^2 + 2x = 9600 \implies x^2 + 2x - 9600 = 0$

Step 3: Factorize

$(x - 96)(x + 100) = 0$

$x = 96 \quad \text{or} \quad x = -100$

Since quantity must be positive ($x > 0$), we reject $x = -100$.

Step 4: Check solution

• Original rate: $\dfrac{2400}{96} = \text{Rs. } 25.00$ per dozen.
• Hypothetical rate: $\dfrac{2400}{98} = \text{Rs. } 24.50$ per dozen.
• Difference: $25.00 - 24.50 = \text{Rs. } 0.50$ $\checkmark$

$\boxed{\text{The man sold } 96 \text{ dozen eggs (at Rs. 25 per dozen)}}$