Skip to content
Numerical Questions · Q1

Q.A and B were partners in a firm sharing profits and losses in the ratio of 3:2. They admit C into the partnership with 1/6 share in the profits. Calculate the new profit sharing ratio?

Bihar BsebTextbookSubjective· 3mImportance★★★★★
48% · 47/97 Questions
✓ Free question

The new profit-sharing ratio after C's admission is 3 : 2 : 1 (A : B : C).

Concept: Admission of a Partner and Reconstitution of Profit-Sharing Ratio

When a new partner is admitted, the existing partnership is reconstituted. The incoming partner acquires their share of profit from the old partners, and we must calculate how the total profit (always taken as 1 or unity) is now divided among all partners.

The treatment depends on what the new partner's share is carved out from:

  1. If the new partner's share is given and the old partners continue in their old ratio (the usual case here): The new partner takes their fraction from the combined share of the old partners, and the old partners sacrifice in their existing ratio.
  2. If the new partner's share and the sacrifice by each old partner are specified separately: We compute each partner's new share individually.

Here, C is admitted for 1/6 share, and A and B were sharing 3:2. The question does not state that A and B will sacrifice in any ratio different from their existing ratio, so A and B will continue to share the remaining profit (after C's share) in their old ratio of 3:2.


Solution

Step 1: Determine the remaining share after C's admission

C's share = 1/6

Remaining share for A and B combined = 1 - 1/6 = 5/6

Step 2: Divide the remaining share between A and B in their old ratio (3:2)

A and B shared profits in the ratio 3:2, which means:

  • A's share in the old firm = 3/5 of the total
  • B's share in the old firm = 2/5 of the total

Now, A and B will share 5/6 (the remaining profit after C's 1/6) in the ratio 3:2.

A's new share = 3/5 × 5/6 = 3/6 = 1/2

B's new share = 2/5 × 5/6 = 2/6 = 1/3

C's share = 1/6 (given)

Step 3: Express the new ratio in simplest whole numbers

The new shares are:

  • A : B : C = 1/2 : 1/3 : 1/6

To convert to whole numbers, find the LCM of the denominators (2, 3, 6), which is 6.

Multiply each fraction by 6:

  • A = 1/2 × 6 = 3
  • B = 1/3 × 6 = 2
  • C = 1/6 × 6 = 1

New profit-sharing ratio = 3 : 2 : 1


Working Note 1: Verification

Total of new ratio = 3 + 2 + 1 = 6

  • A's share = 3/6 = 1/2 ✓
  • B's share = 2/6 = 1/3 ✓
  • C's share = 1/6 ✓

Sum = 1/2 + 1/3 + 1/6 = (3 + 2 + 1)/6 = 1 ✓


Tip

When the new partner's share is given and nothing else is specified, always assume the old partners sacrifice in their existing ratio. The remaining profit after the new partner's share is divided among the old partners in their old ratio.

Watch out

A common mistake is to simply add C's share to the old ratio (e.g., writing 3:2:1 without calculation). This is incorrect because the old partners' absolute shares must reduce to accommodate the new partner. Always calculate the remaining share first, then redistribute it.


✓Final answer

After admitting C with a 1/6 share, and A and B continuing to share the remaining profit in their old ratio of 3:2, the new profit-sharing ratio is 3 : 2 : 1 (A : B : C).

Unlock everything free for 14 days

  • Full step-by-step solutions
  • Concept-first explanations
  • Methods, shortcuts & mistakes
  • PYQ mapping + timed mock tests

Full access for 14 days. No credit card required.