Explanation:
To find the sacrificing ratio, we need to understand the concept of sacrificing ratio. The sacrificing ratio is the ratio in which the existing partners give up their share of profits in order to admit a new partner. It is calculated by comparing the old profit sharing ratio with the new profit sharing ratio.

Given:
Old profit sharing ratio of A and B = 2 : 1
New profit sharing ratio = 3 : 1 : 1

Step 1: Determine the share of A and B in the new ratio
In the new ratio, the share of A and B is 3 out of 5 parts (3 + 1 + 1 = 5).
So, the share of A = (3/5) * 1/2 = 3/10
And the share of B = (3/5) * 1/2 = 3/10

Step 2: Calculate the difference in share for A and B
The difference in share for A = (Old share of A) - (New share of A)
= 1/2 - 3/10
= 5/10 - 3/10
= 2/10
= 1/5

The difference in share for B = (Old share of B) - (New share of B)
= 1/2 - 3/10
= 5/10 - 3/10
= 2/10
= 1/5

Step 3: Calculate the sacrificing ratio
The sacrificing ratio is the ratio in which A and B give up their shares. Since both A and B give up their shares in the same ratio, the sacrificing ratio will be the same for both of them.
Therefore, the sacrificing ratio is 1/5 : 1/5, which can be simplified to 1 : 1/5.

Step 4: Simplify the sacrificing ratio
To simplify the sacrificing ratio, we need to convert the fractional part to a whole number. We can do this by multiplying both the numerator and the denominator by the same number.

In this case, we can multiply both the numerator and the denominator by 5 to get rid of the fractional part.

Multiplying 1 by 5, we get 5.
Multiplying 1/5 by 5, we get 1.

So, the sacrificing ratio is 5 : 1.

Step 5: Choose the correct option
Now, we need to compare the sacrificing ratio we calculated with the given options.

Option a) 2 : 1 - This is not the correct ratio.
Option b) 1 : 7 - This is not the correct ratio.
Option c) 3 : 5 - This is not the correct ratio.
Option d) 1 : 2 - This is the correct ratio.

Therefore, the correct answer is option d) 1 : 2.