Solution:

Given, X, Y, Z are equal partners in a firm. Z retires from the firm. The new profit sharing ratio between X and Y is 1:2.

We need to find the gaining ratio.

Gaining Ratio:

The ratio in which the continuing partners (X and Y) share the profits of the partnership firm due to the retirement or death of any partner is called the gaining ratio.

Let the gaining ratio be x:y.

After Z's retirement, the new ratio of sharing profits between X and Y is 1:2. So, the total profit is divided into 3 parts, out of which Y gets 2 parts and X gets 1 part.

Therefore, the new profit sharing ratio of X and Y is:

X:Y = 1:2

Let us assume that the total profit is 3 units.

So, Y will get 2 units and X will get 1 unit.

Old Profit Sharing Ratio:

Before Z's retirement, the profit was divided equally among the three partners X, Y, and Z.

So, the old profit sharing ratio of X, Y, and Z is:

X:Y:Z = 1:1:1

Let us assume that the old profit is 3 units.

So, each partner gets 1 unit.

Calculation of Gaining Ratio:

Now, Z retires from the firm. So, the total profit is shared only between X and Y. Z will not get any share from the new profit.

The difference in the profit share of X and Y is the gain or loss for each partner.

In this case, Y gains more profit than X. So, the gaining ratio will be in favor of Y.

Let us calculate the profit gained by Y and X after Z's retirement.

Profit gained by Y:

Y's new share - Y's old share

= 2 units - 1 unit

= 1 unit

Profit gained by X:

X's new share - X's old share

= 1 unit - 1 unit

= 0 unit

Therefore, the gaining ratio of Y and X is 1:0.

Simplifying the ratio, we get:

1:0 = 4:0 (multiplying both sides by 4)

= 4:1

Hence, the gaining ratio is 4:1.

Option (c) is incorrect as it says the gaining ratio is 2:1, which is not true.

Option (a) is incorrect as it says the gaining ratio is 3:2, which is not true.

Option (b) is incorrect as it says only B gains by 1/3, which is not true.