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Numerical Questions · Q5

Q.P and Q are partners sharing profits in 2:1 ratio. They admitted R into partnership giving him 1/5 share which he acquired from P and Q in 1:2 ratio. Calculate new profit sharing ratio?

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The new profit-sharing ratio after R's admission is 3:1:1 (P:Q:R).

Concept: Admission of a Partner and Sacrifice Ratio

When a new partner is admitted, the old partners sacrifice a portion of their share in favour of the incoming partner. The new profit-sharing ratio is calculated by deducting each old partner's sacrifice from their old share.

The question tells us two things:

  1. R acquires a 1/5 share in total.
  2. This 1/5 is acquired from P and Q in the ratio 1:2.

The phrase "acquired from P and Q in 1:2 ratio" means P and Q sacrifice in the ratio 1:2. So out of the 1/5 share given to R, P sacrifices 1/3 of it and Q sacrifices 2/3 of it.

Watch out

A common mistake is to assume R's share comes equally from both partners or in their old profit-sharing ratio. Always read carefully how the new partner's share is acquired — the sacrifice ratio may differ from the old profit-sharing ratio.

Solution

Step 1: Calculate the sacrifice of each old partner

R's total share = 1/5

P and Q sacrifice this in the ratio 1:2.

  • P's sacrifice = 1/5 × 1/(1+2) = 1/5 × 1/3 = 1/15
  • Q's sacrifice = 1/5 × 2/(1+2) = 1/5 × 2/3 = 2/15

Step 2: Calculate the new shares of P and Q

Old ratio of P and Q = 2:1

Converting to fractions with a common denominator:

  • P's old share = 2/3
  • Q's old share = 1/3

New shares:

  • P's new share = Old share − Sacrifice = 2/3 - 1/15 = 10/15 - 1/15 = 9/15 = 3/5
  • Q's new share = Old share − Sacrifice = 1/3 - 2/15 = 5/15 - 2/15 = 3/15 = 1/5
  • R's new share = 1/5

Step 3: Express the new ratio

New profit-sharing ratio:

  • P : Q : R = 3/5 : 1/5 : 1/5

Multiplying throughout by 5 to remove fractions: …

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