Q.P, Q and R are partners sharing profits in the ratio of 5:3:2. They admit S as a new partner, and it is agreed that the new profit-sharing ratio of P, Q, R and S will be 4:3:2:1. Calculate the sacrificing ratio.

Here the new ratio is given directly for all four partners, so the sacrificing ratio must be worked out by comparing each partner's old share to their new share individually — it need not match the old ratio at all.

Step 1 — Express old and new shares over the same denominator

Old ratio P:Q:R = 5:3:2 (out of 10), so P = 5/10, Q = 3/10, R = 2/10.

New ratio P:Q:R:S = 4:3:2:1 (out of 10), so P = 4/10, Q = 3/10, R = 2/10, S = 1/10.

Step 2 — Compute each old partner's sacrifice

Check: total sacrifice = 1/10, which exactly equals S's new share of 1/10 ✓ — confirming S's entire share has come from a single partner, P.

Step 3 — State the Sacrificing Ratio

Since only P has given up any share, and Q's and R's shares are completely unchanged, the sacrifice is made by P alone. The Sacrificing Ratio is therefore P alone (or, expressed among the three, P:Q:R = 1:0:0) — the entire premium for goodwill that S brings in will be credited only to P's capital account, and neither Q nor R will receive any share of it. …

A frequent mistake is assuming that because three old partners exist, all three must share in the sacrifice — always check each partner's individual Old Share − New Share; a partner whose share is unchanged (or has increased) contributes nothing t …