Financial Accounting · Admission of Partner

New Profit Sharing Ratio and Sacrificing Ratio on Admission

Updated 10 October 2026 · Fact-checked

When a partner is admitted, old partners give up part of their profit share. The sacrificing ratio is old share minus new share for each old partner. The new ratio is each old partner's share after the sacrifice, plus the new partner's share. Always check that the new shares add up to 1.

Understand New Profit Sharing Ratio and Sacrificing Ratio

A partnership firm runs on a profit sharing ratio. When a new partner joins, the old ratio ends and a new one starts. The new partner must get a share, and that share has to come from somewhere. It comes from the old partners.

The new profit sharing ratio is the ratio in which all partners, old and new, share profits after admission. The sacrificing ratio is the ratio in which the old partners give up their shares. It matters because the new partner pays for goodwill, and the old partners who sacrificed are the ones compensated.

The sacrifice of each old partner is: old share − new share. A positive answer means a sacrifice. A negative answer means that partner has gained, which is the gaining ratio. On admission this is possible only when the agreement gives an old partner a larger share than before.

The question always tells you how the new partner's share is taken. There are three common cases: the new partner takes an agreed share from the old partners in their old ratio; takes it in a stated proportion (for example, A gives 2/3 and B gives 1/3 of it); or the new ratios are given directly. Each case needs a slightly different first step, but the method after that is the same.

So, for any question, fix the new share of every partner as a fraction of 1, subtract from the old shares, and read off the sacrifice.

Key rules to remember

Sacrifice of an old partner: Sacrifice = Old share − New share; A positive value is a sacrifice. A negative value is a gain.
Sacrificing ratio: Sacrificing ratio = Sacrifice of A : Sacrifice of B : ...; Simplify the fractions to a common denominator and then compare the numerators.
Case 1: new partner's share taken in old ratio: New share of old partner = Old share × (1 − New partner's share); The sacrificing ratio then equals the old ratio. Each partner's sacrifice = Old share × New partner's share.
Case 2: share taken in a stated proportion: Sacrifice of A = New partner's share × A's stated fraction of sacrifice; New share of A = Old share of A − Sacrifice of A.
Check: Sum of all new shares = 1; Also check that the total sacrifice equals the new partner's share.
Gaining ratio (contrast): Gain = New share − Old share; Used when a partner retires or when an old partner increases his share. It is the opposite of sacrifice.

How to solve New Profit Sharing Ratio and Sacrificing Ratio questions

Use this method for any question on new ratio and sacrificing ratio at admission.

1Write the old ratio of the partners as fractions of 1.
2Write the new partner's share and note how it is to be acquired: from old partners in the old ratio, in a stated proportion, or as a given new ratio.
3Work out the sacrifice of each old partner, or the new share directly if the new ratios are given.
4Subtract the sacrifice from the old share to get each old partner's new share, or calculate new share = old share − sacrifice.
5Write the new ratio with a common denominator, and check that the shares total 1.
6If the new ratio was given, find the sacrifice as old share − new share for each old partner. Compare the numerators to state the sacrificing ratio.
7State both ratios clearly, with the working shown as fractions.

Quickest way: Common denominator shortcut

When to use it: Use this when the question gives old ratios and a new partner's fraction and you need both ratios fast.

Convert old shares and the new partner's share to fractions with the same denominator.
If the share is taken in the old ratio, the sacrificing ratio is the old ratio. Do not calculate further for that part.
If a stated proportion is given, multiply the new partner's share by each fraction to get the sacrifices.
Subtract to find the new shares. The numerators over the common denominator give the new ratio.
Add the numerators of new shares. The total must equal the denominator.

Common mistakes in New Profit Sharing Ratio and Sacrificing Ratio

Writing the new ratio as the old ratio plus the new partner's share.
Students add the fractions directly instead of reducing the old shares first.
Fix: The old partners' shares fall by the sacrifice. Subtract first, then add the new partner's share. Check that the total is 1.
Calculating sacrifice as new share − old share.
It is the reverse of the formula and mixes up sacrifice and gain.
Fix: Sacrifice = Old − New. If the answer is negative, the partner has gained.
Assuming the sacrificing ratio always equals the old ratio.
It holds only when the new partner takes his share in the old ratio.
Fix: Read the question. If the sacrifice is in a stated proportion or the new ratio is given, calculate it separately.
Comparing fractions with different denominators.
Students compare 1/2 and 3/10 and state the ratio as 1:3 by mistake.
Fix: Convert all fractions to a common denominator first, then use the numerators.
Confusing the sacrificing ratio with the gaining ratio.
Both use old and new shares, and the formulas look alike.
Fix: Sacrificing ratio applies to partners who give up share, usually at admission. Gaining ratio applies to partners who take more share, usually at retirement. Goodwill goes from gainers to sacrificers.
Not stating the ratios in simplest form.
Students stop at fractions with large denominators.
Fix: Reduce the ratio by the common factor and write it as, for example, 3 : 2.

Worked examples

Example 1

A and B share profits in the ratio 3 : 2. They admit C for a 1/5 share in profits. C gets this share from A and B in their old ratio. Find the new profit sharing ratio and the sacrificing ratio.

Show the solution

Old shares: A = 3/5, B = 2/5.
C's share = 1/5. It is taken in the old ratio 3 : 2.
Sacrifice of A = 1/5 × 3/5 = 3/25. Sacrifice of B = 1/5 × 2/5 = 2/25.
Total sacrifice = 3/25 + 2/25 = 5/25 = 1/5, which equals C's share.
New share of A = 3/5 − 3/25 = 15/25 − 3/25 = 12/25.
New share of B = 2/5 − 2/25 = 10/25 − 2/25 = 8/25.
C = 1/5 = 5/25.
Check: 12 + 8 + 5 = 25, so the total is 1.
Sacrificing ratio = 3 : 2, the same as the old ratio.

Answer: New ratio A : B : C = 12 : 8 : 5. Sacrificing ratio A : B = 3 : 2.

Example 2

X and Y share profits in the ratio 5 : 3. They admit Z for a 1/4 share. X gives up 2/3 of Z's share and Y gives up 1/3 of it. Find the new profit sharing ratio and the sacrificing ratio. Then, if the new ratio had instead been given as X : Y : Z = 3 : 3 : 2, state the sacrifice or gain of X and Y.

Show the solution

Old shares: X = 5/8, Y = 3/8.
Z's share = 1/4.
Sacrifice of X = 1/4 × 2/3 = 2/12 = 1/6.
Sacrifice of Y = 1/4 × 1/3 = 1/12.
Total sacrifice = 1/6 + 1/12 = 2/12 + 1/12 = 3/12 = 1/4, which equals Z's share.
New share of X = 5/8 − 1/6 = 15/24 − 4/24 = 11/24.
New share of Y = 3/8 − 1/12 = 9/24 − 2/24 = 7/24.
Z = 1/4 = 6/24.
Check: 11 + 7 + 6 = 24, so the total is 1.
Sacrificing ratio = 1/6 : 1/12 = 2/12 : 1/12 = 2 : 1.
For the second part, the new ratio is 3 : 3 : 2 with total 8. X = 3/8, Y = 3/8, Z = 2/8.
Sacrifice of X = 5/8 − 3/8 = 2/8. Sacrifice of Y = 3/8 − 3/8 = 0.
Total sacrifice = 2/8 = Z's share, so the working is consistent.

Answer: First part: new ratio X : Y : Z = 11 : 7 : 6 and sacrificing ratio X : Y = 2 : 1. Second part: X sacrifices 2/8 (1/4) and Y sacrifices nothing, so Z's whole share comes from X.

Exam tips

Write every share as a fraction, not a decimal. Fractions let you check that the total is 1.
Read the sentence on how the share is acquired. It decides whether the sacrificing ratio equals the old ratio.
Show the check line (total = 1) in written answers. It earns marks and catches arithmetic slips.
In MCQs, test the options by adding the shares. Wrong options often fail to total 1.
Carry the sacrificing ratio forward. The next step in the same question usually shares goodwill in that ratio.

Practice questions from Admission of Partner

New Profit Sharing Ratio and Sacrificing Ratio: frequently asked questions

What is the sacrificing ratio formula?

Sacrifice of a partner = Old share − New share. The sacrificing ratio is the ratio of these sacrifices among the old partners. Use a common denominator before you compare them.

How do you calculate the new profit sharing ratio on admission?

Find how much each old partner gives up. Subtract it from the old share. Add the new partner's share. Express all shares over a common denominator and write the numerators as the ratio.

What is the difference between sacrificing ratio and gaining ratio?

Sacrificing ratio is old share minus new share for partners who give up profit, usually on admission. Gaining ratio is new share minus old share for partners who take more profit, usually on retirement. Goodwill compensation flows from gainers to sacrificers.

When is the sacrificing ratio equal to the old ratio?

It equals the old ratio when the new partner takes his share from the old partners in their old ratio and nothing else is stated. If the question gives another proportion or a new ratio, calculate it separately.