Q.P and Q share profits in the ratio 5:3. They admit R, who acquires his share of 1/4 from P and Q in the ratio 3:1. Find the new profit-sharing ratio and the sacrificing ratio.

R's 1/4 share is split 3:1 between P and Q, so P sacrifices 3/16 and Q sacrifices 1/16 of the whole.

P's old share 5/8 = 10/16, less 3/16 sacrificed = 7/16. Q's old share 3/8 = 6/16, less 1/16 sacrificed = 5/16. R's share = 4/16.

R's total share is 1/4, acquired from P and Q in the ratio 3:1, so R takes 3/4 of 1/4 = 3/16 from P, and 1/4 of 1/4 = 1/16 from Q.

Converting P and Q's old ratio (5:3, i.e., P = 5/8 and Q = 3/8) to sixteenths: P = 5/8 = 10/16, Q = 3/8 = 6/16.

P's new share = 10/16 − 3/16 = 7/16

Q's new share = 6/16 − 1/16 = 5/16

R's share = 3/16 + 1/16 = 4/16

New ratio P : Q : R = 7 : 5 : 4 (out of 16).

Check: 7/16 + 5/16 + 4/16 = 16/16 = 1, confirming the split is complete and consistent.

Sacrificing ratio: P's sacrifice = 10/16 − 7/16 = 3/16; Q's sacrifice = 6/16 − 5/16 = 1/16. Sacrificing ratio of P and Q = 3/16 : 1/16 = 3 : 1 — which, as it should, exactly matches the 3:1 ratio in which R was stated to have acquired his share.

Students sometimes apply the 3:1 acquisition ratio directly to P and Q's OLD shares (5 and 3) instead of to R's own 1/4 share — the 3:1 split describes how R's incoming share is sourced, not a re-division of P and Q's existing shares.