Q.X and Y are partners sharing profits in the ratio of 5:3. They admit Z into partnership. Z acquires his share of 3/10th from X and Y in the ratio of 2:1. Calculate the new profit-sharing ratio and the sacrificing ratio.

Here Z's share is not given directly as a carve-out of the whole — instead Z takes a specified 3/10th share FROM X and Y, split 2:1 between them, so each old partner's new share is found by direct subtraction.

Step 1 — Find how much each old partner gives up

Z's total share = 3/10, acquired from X and Y in the ratio 2:1.

X's sacrifice = 2/3 × 3/10 = 6/30 = 1/5

Y's sacrifice = 1/3 × 3/10 = 3/30 = 1/10

Check: 1/5 + 1/10 = 2/10 + 1/10 = 3/10 = Z's total share ✓

Step 2 — Compute each old partner's new share

Using a common denominator of 40:

X's old share = 5/8 = 25/40; X's sacrifice = 1/5 = 8/40 → X's new share = 25/40 − 8/40 = 17/40

Y's old share = 3/8 = 15/40; Y's sacrifice = 1/10 = 4/40 → Y's new share = 15/40 − 4/40 = 11/40

Z's share = 3/10 = 12/40

Check: 17/40 + 11/40 + 12/40 = 40/40 = 1 ✓

New Profit-Sharing Ratio, X : Y : Z = 17 : 11 : 12

Step 3 — Sacrificing Ratio …

As a shortcut, whenever a question states the ratio in which the new partner ACQUIRES their share from the old partners, that stated ratio IS the sacrificing ratio directly — there is no need to separately com …

A common error is subtracting the WHOLE of Z's share (3/10) from each of X and Y instead of splitting it 2:1 between them first. Always convert all fractions t …