Step-by-Step Solution

Step-by-Step Solution

Step 1: Define variables and formulate rate equation

Let $x$ be the number of sheep bought ($x > 0$).

• Original rate per sheep $= \dfrac{9000}{x}$
• Hypothetical rate per sheep $= \dfrac{9000}{x + 3}$

Given that the price difference is Rs. 100: $\frac{9000}{x} - \frac{9000}{x + 3} = 100$

Step 2: Simplify and form the quadratic equation

Divide through by $100$: $\frac{90}{x} - \frac{90}{x + 3} = 1$

Combine fractions over $x(x + 3)$: $90 \left[\frac{(x + 3) - x}{x(x + 3)}\right] = 1$

$\frac{270}{x^2 + 3x} = 1 \implies x^2 + 3x - 270 = 0$

Step 3: Factorize

$(x - 15)(x + 18) = 0$

$x = 15 \quad \text{or} \quad x = -18$

Since sheep count must be positive ($x > 0$), we reject $x = -18$.

Step 4: Check solution

• Original rate: $\dfrac{9000}{15} = \text{Rs. } 600$ per sheep.
• Slower/hypothetical rate: $\dfrac{9000}{18} = \text{Rs. } 500$ per sheep.
• Difference: $600 - 500 = \text{Rs. } 100$ $\checkmark$

$\boxed{\text{The farmer bought } 15 \text{ sheep (at Rs. 600 each)}}$