Sarvy invested $$ \frac{1}{3} $$rd of her capital at 10% gain, $$ \frac{1}{4} $$th at 20% gain, and the remaining at 30% gain. If her income from gain is ₹50,000, then the value of her capital is:

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Let Sarvy's total capital be $$ x $$. The income from each part of the investment can be calculated as follows:

1. **First part ($$ \frac{1}{3} $$rd at 10% gain):**  
   Income = $$ \frac{x}{3} 	imes \frac{10}{100} = \frac{x}{30} $$.

2. **Second part ($$ \frac{1}{4} $$th at 20% gain):**  
   Income = $$ \frac{x}{4} 	imes \frac{20}{100} = \frac{x}{20} $$.

3. **Remaining part ($$ x - (\frac{x}{3} + \frac{x}{4}) $$ at 30% gain):**  
   Remaining capital = $$ x - (\frac{x}{3} + \frac{x}{4}) = x - \frac{7x}{12} = \frac{5x}{12} $$.  
   Income = $$ \frac{5x}{12} 	imes \frac{30}{100} = \frac{15x}{120} = \frac{x}{8} $$.

Adding all incomes:  
$$ 
	ext{Total income} = \frac{x}{30} + \frac{x}{20} + \frac{x}{8}. 
$$  
Taking the LCM (120):  
$$ 
	ext{Total income} = \frac{4x}{120} + \frac{6x}{120} + \frac{15x}{120} = \frac{25x}{120}. 
$$

Given that total income is ₹50,000:  
$$ 
\frac{25x}{120} = 50000. 
$$  
Solving for $$ x $$:  
$$ 
x = 50000 	imes \frac{120}{25} = 240000. 
$$

Thus, the value of Sarvy's capital is **₹2,40,000**, corresponding to option (b).

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