CAT 2024 Actual Questions Equations

Detailed Solution:
Step 1: Understand the problem
Let the initial quantity of grains be \( x \) kg.
Step 2: Simplify after the first customer
The first customer takes \( \frac{x}{2} + 3 \):
Remaining grains = \( x - \left(\frac{x}{2} + 3\right) = \frac{x}{2} - 3 \).

Step 3: Simplify after the second customer
The second customer takes \( \frac{1}{2}\left(\frac{x}{2} - 3\right) + 3 = \frac{x}{4} + \frac{3}{2} \).
Remaining grains = \( \frac{x}{2} - 3 - \left(\frac{x}{4} + \frac{3}{2}\right) = \frac{x}{4} - \frac{9}{2} \).

Step 4: Simplify after the third customer
The third customer takes \( \frac{1}{2}\left(\frac{x}{4} - \frac{9}{2}\right) + 3 = \frac{x}{8} + \frac{3}{4} \).
Remaining grains = \( \frac{x}{4} - \frac{9}{2} - \left(\frac{x}{8} + \frac{3}{4}\right) = 0 \).

Step 5: Solve for \( x \)
Solve \( \frac{x}{8} - \frac{21}{4} = 0 \):
Multiply through by 8:
\( x = 42 \).

Final Answer: The initial quantity of grains is \( \boxed{42} \).