New, sacrificing & gaining ratio
Pick admission, retirement or death, choose how the share changes hands, and get the new profit-sharing ratio with every fraction worked out — no rounding.
C gets 1/5 of the firm. The old partners give it up in the ratio 3 : 2.
Juice Shop Partners
Watch each partner’s slice pour into C’s glass.
- A3/5
- B2/5
- C0
- A12/25
- B8/25
- C1/5
| Partner | Old | New | Sacrifice / Gain |
|---|---|---|---|
| A | 3/5 | 12/25 | Sacrifice 3/25 |
| B | 2/5 | 8/25 | Sacrifice 2/25 |
| C | 0 | 1/5 | Gain 1/5 |
Working
- Old shares from 3:2: A 3/5, B 2/5.
- C’s share is 1/5. A and B give it up in their old ratio 3:2, so each keeps 4/5 of his old share.
- A’s new share = 3/5 × 4/5 = 12/25.
- B’s new share = 2/5 × 4/5 = 8/25.
- A: 3/5 − 12/25 = 15/25 − 12/25 = 3/25
- B: 2/5 − 8/25 = 10/25 − 8/25 = 2/25
- Check: 12/25 + 8/25 + 5/25 = 1
- Sacrificing ratio = 3:2 (the same as the old ratio 3:2).
- New ratio A:B:C = 12:8:5.
Common mistakes in the exam
- “1/3 of his share” means 1/3 of his own old share. “1/3 share” means a 1/3 share of the whole firm. Read the question carefully.
- The sacrificing ratio is not always the old ratio. It is the old ratio only when the new partner’s share comes from the old partners in their old ratio.
- In retirement, the gaining ratio is new minus old for each continuing partner, not the new ratio itself.
- Check that the new shares add up to 1 before you write the ratio.
Three ratios, one idea
The new ratio is the profit-sharing ratio after the change. The sacrificing ratio is the ratio in which the old partners gave up their shares. The gaining ratio is the ratio in which the continuing partners gained when someone left. Each one is worked out from the change in shares, so always find the shares first, then the ratio.
Admission: five ways the share can be given up
- In the old ratio. The new partner’s share is taken from the old partners in the ratio they already share in. Their sacrifice is in the old ratio too.
- Own share. Each old partner gives up a fraction of his own old share. Multiply his old share by that fraction.
- Firm share. Each old partner gives up a fraction of the whole firm. Subtract it straight from his old share.
- Equal. The new partner’s share is given up in equal parts. Divide it by the number of old partners.
- New ratio given. The full new ratio, including the new partner, is stated. Each old partner’s sacrifice is his old share minus his new share.
Retirement and death: three ways
The retiring partner’s whole share goes to the partners who stay. If nothing is said, they share it in their old ratio. If a new ratio is given, the gain is the new share minus the old share. If a ratio is stated for the retiring share, each partner gets their part of it.
Check your answer
Add up the new shares: they must make exactly 1. Write the ratio in lowest whole numbers. A negative sacrifice means a partner gains in an admission, which is unusual, so check the figures again.
Once the new ratio is correct, the next question is usually what the new partner pays for goodwill. Next: value the goodwill.
Learn the chapter
Learning aid — check the method with your textbook and teacher. Follow the wording of your question: if it states a ratio or a share, use that.
By the ACES team · Last checked 6 November 2026