Q.A and B are partners sharing profits and losses in the ratio of 3:2. They admit C into partnership, giving C a 1/5th share of profits. Nothing else is stated about how A and B will share the remaining profit. Calculate the new profit-sharing ratio and the sacrificing ratio.

Since only C's share is given and nothing more is stated about A and B, the remaining share is split between them in their OLD ratio of 3:2.

Remaining share for A and B together = 1 − 1/5 = 4/5, split 3:2 between them.

Step 1 — Note C's share and the remaining share

C's share = 1/5

Remaining share (for A and B together) = 1 − 1/5 = 4/5

Since nothing else is stated, A and B continue to share this remaining 4/5 in their OLD ratio of 3:2.

Step 2 — Compute A's and B's new shares

A's new share = 3/5 × 4/5 = 12/25

B's new share = 2/5 × 4/5 = 8/25

C's share = 1/5 = 5/25

Check: 12/25 + 8/25 + 5/25 = 25/25 = 1 ✓

New Profit-Sharing Ratio, A : B : C = 12 : 8 : 5

Step 3 — Compute the Sacrificing Ratio

Sacrificing Ratio, A : B = 3 : 2

Note that this sacrificing ratio is identical to the old ratio (3:2) — this is exactly what happens whenever the new partner's share is deducted from the remaining share in the old partners' existing ratio, as confirmed in the concept note above.

Alternative check: since the remainder is shared in the old ratio, the sacrificing ratio can be found directly without computing new shares first — it will always equal the old ratio (3:2) in this specific case, because each old partner's share is being scaled down by the same fraction (4/5).

A frequent error is dividing C's share equally between A and B by default — the correct default (when nothing else is stated) is to split the remainder in the OLD ratio, not equally. Also check that the three new shares add up to exactly 1 (or the chosen common denominator) before finalising the ratio.