**Calculation of Gaining Ratio:**

Gaining ratio is the ratio in which the remaining partners acquire the share of a retiring partner. In this case, when partner C retires, his share is acquired by partners A and B in the ratio of 6:4.

To calculate the gaining ratio, we need to find the ratio of the new investments made by partners A and B to the total investment of partner C.

Partner A invests Rs. 1,00,000 and Partner B invests Rs. 75,000. Therefore, the new investment made by partners A and B is Rs. 1,00,000 + Rs. 75,000 = Rs. 1,75,000.

Partner C's total investment is Rs. 50,000.

Gaining Ratio = (New Investment of A + New Investment of B) / Total Investment of C
= Rs. 1,75,000 / Rs. 50,000
= 3.5

Therefore, the gaining ratio is 3.5:1.

**Calculation of New Profit Sharing Ratio:**

The new profit sharing ratio is the ratio in which the profits will be distributed among the remaining partners after the retirement of partner C.

The new profit sharing ratio is calculated by adding the gaining ratio to the existing profit sharing ratio.

The existing profit sharing ratio is given as A:B:C = 1,00,000:75,000:50,000 = 2:1:1

Therefore, the new profit sharing ratio is A:B:C = 2 + 3.5 : 1 + 3.5 : 1 = 5.5:4.5:1

Hence, the new profit sharing ratio is 5.5:4.5:1.

**Summary:**

- Gaining Ratio: 3.5:1
- New Profit Sharing Ratio: 5.5:4.5:1

Partner C's share is acquired by partners A and B in the ratio of 6:4, resulting in a gaining ratio of 3.5:1. The new profit sharing ratio after the retirement of partner C is 5.5:4.5:1, which is calculated by adding the gaining ratio to the existing profit sharing ratio.