Let's analyze the given profit sharing ratio of the partners L, S, and D.

The given ratios are \( \frac{2}{5} \), \( \frac{3}{7} \), and \( \frac{3}{10} \).

To find the combined ratio, we need a common denominator. The least common multiple (LCM) of 5, 7, and 10 is 70.

• L's share: \( \frac{2}{5} = \frac{2 \times 14}{5 \times 14} = \frac{28}{70} \)
• S's share: \( \frac{3}{7} = \frac{3 \times 10}{7 \times 10} = \frac{30}{70} \)
• D's share: \( \frac{3}{10} = \frac{3 \times 7}{10 \times 7} = \frac{21}{70} \)

The original profit sharing ratio of L, S, and D is 28:30:21.

Calculating Gain Ratio Upon Retirement

Partner S retires from the firm. The continuing partners are L and D.

When a partner retires, the continuing partners usually gain a portion of the retiring partner's share. The ratio in which they acquire this share is called the gain ratio.

In the absence of any specific agreement on how the retiring partner's share is acquired, the continuing partners acquire the share in their old profit sharing ratio.

The old profit sharing ratio between L and D was 28:21 (from the original ratio 28:30:21).

Let's simplify the ratio 28:21 by dividing both numbers by their greatest common divisor, which is 7.

• 28 </ 7 = 4
• 21 </ 7 = 3

So, the old ratio between L and D was 4:3.

Assuming L and D acquire S's share in their old profit sharing ratio, their gain ratio will be 4:3.

Understanding the Gain Ratio of L

The question asks for the "gain ratio of L". This usually refers to L's proportion in the total gain ratio of the continuing partners.

The gain ratio of L and D is 4:3.

Total of the gain ratio = 4 + 3 = 7.

L's proportion of the gain = \( \frac{\text{L's share in gain ratio}}{\text{Total gain ratio}} = \frac{4}{7} \).

Now let's compare this calculated proportion with the given options:

• Option 1: \( \frac{1}{4} \)
• Option 2: \( \frac{1}{3} \)
• Option 3: \( \frac{28}{70} = \frac{2}{5} \)
• Option 4: More than one of the above
• Option 5: None of the above

Our calculated gain proportion for L is \( \frac{4}{7} \).

\( \frac{4}{7} \) is not equal to \( \frac{1}{4} \).

\( \frac{4}{7} \) is not equal to \( \frac{1}{3} \).

\( \frac{4}{7} \) is not equal to \( \frac{2}{5} \).

Since none of the options match the calculated gain proportion of L, the correct answer is None of the above.