A:B::3:1.C is admitted for 1÷4th share. What is the sacrificing ratio?

 A:B::3:1.C is admitted for 1÷4th share. What is the sacrificing ratio?

To determine the sacrificing ratio, we need to calculate the ratio in which A and B are sacrificing their share in the partnership to accommodate C's admission. 

Given: A:B = 3:1

Let's assume A sacrifices 'x' and B sacrifices 'y' from their respective shares. 

So, A's new share = A's original share - A's sacrifice = 3 - x

And B's new share = B's original share - B's sacrifice = 1 - y

According to the given information, C is admitted for a 1/4th share. This means C's share is 1/4 or (1/4) * (total share).

Since the total share is the sum of the new shares of A, B, and C, we can write the equation:

A's new share + B's new share + C's share = Total share

(3 - x) + (1 - y) + (1/4) = 1

Simplifying the equation, we have:

4 - x - y + 1/4 = 1

4 - x - y = 1 - 1/4

4 - x - y = 3/4

To find the sacrificing ratio, we compare the sacrifices made by A and B. So, we equate their sacrifices:

x/y = A's original share/B's original share

3/1 = x/y

Cross-multiplying, we get:

3y = x

Substituting the value of x in the equation 4 - x - y = 3/4, we have:

4 - 3y - y = 3/4

4 - 4y = 3/4

16 - 16y = 3

16y = 13

y = 13/16

Now, substituting the value of y in the equation x = 3y, we get:

x = 3 * (13/16)

x = 39/16

Therefore, the sacrificing ratio is:

A:B = x:y = (39/16):(13/16) = 39:13

The sacrificing ratio between A and B is 39:13.