X,Y and Z Old ratio=1/4 : 2/5 : 7/20 = 5/20 : 8/20 : 7/20 = 5 : 8 : 7. if Z retired, X and Y new ratio = 5 : 8

Calculation of New Profit-Sharing Ratio

To calculate the new profit-sharing ratio, we need to consider the ratio in which the partners were sharing profits initially and the changes that occur after Z's retirement.

Initial Profit-Sharing Ratio:
X : Y : Z = 1/4 : 2/5 : 7/20

Step 1: Convert the ratios to a common denominator.
In this case, the common denominator is 20.

X : Y : Z = 5/20 : 8/20 : 7/20

Step 2: Simplify the ratios.
X : Y : Z = 5 : 8 : 7

Step 3: Calculate the total ratio.
The total ratio represents the sum of individual ratios and helps us determine the share of each partner.

Total ratio = 5 + 8 + 7 = 20

Step 4: Calculate the share of each partner.
To find the share of each partner, divide the total ratio by the individual ratio.

X's share = Total ratio/Individual ratio = 20/5 = 4
Y's share = Total ratio/Individual ratio = 20/8 = 2.5
Z's share = Total ratio/Individual ratio = 20/7 = 2.857

Step 5: Adjust the remaining share after Z's retirement.
Since Z retires from the firm, his share needs to be distributed among the remaining partners. In this case, X and Y will share Z's portion.

Total share of X and Y = X's share + Y's share = 4 + 2.5 = 6.5

Step 6: Calculate the new profit-sharing ratio.
The new profit-sharing ratio is calculated by dividing the individual share of each partner by the total share of X and Y.

X's new share = X's share/Total share of X and Y = 4/6.5 ≈ 0.615
Y's new share = Y's share/Total share of X and Y = 2.5/6.5 ≈ 0.385

Therefore, the new profit-sharing ratio after Z's retirement is approximately 0.615 : 0.385, which can be simplified as 61.5 : 38.5.