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Q.P, Q and R were partners in a firm sharing profits and losses in the ratio of 6 : 5 : 4. They admitted S as a new partner for 1/8th share in the profits of the firm. It was agreed that Q would retain his original share. The sacrificing ratio of P and R will be : (A) 6 : 5 (B) 4 : 5 (C) 3 : 2 (D) 5 : 4

CBSECBSE Class XII Board 2026MCQ· 1mImportance★★★★★
✓ Free question

The sacrificing ratio of P and R is 3 : 2 (Option C).

Concept: Sacrificing Ratio on Admission of a Partner

When a new partner is admitted, the existing partners sacrifice a portion of their profit share in favour of the incoming partner. The sacrificing ratio measures how much each old partner gives up. It is calculated as:

Sacrificing Ratio=Old Share−New Share\text{Sacrificing Ratio} = \text{Old Share} - \text{New Share}

The partner who sacrifices more is entitled to a larger share of goodwill compensation from the new partner. In this problem, Q retains his original share (sacrifices nothing), so only P and R sacrifice. We must first determine the new profit-sharing ratio, then compute individual sacrifices.

Treatment and Solution

Step 1: Determine the old profit-sharing ratio

P, Q and R share profits in the ratio 6:5:46 : 5 : 4.

Total parts = 6+5+4=156 + 5 + 4 = 15

  • P's old share = 615=25\frac{6}{15} = \frac{2}{5}
  • Q's old share = 515=13\frac{5}{15} = \frac{1}{3}
  • R's old share = 415\frac{4}{15}

Step 2: S is admitted for 18\frac{1}{8} share

The remaining share available for P, Q and R together = 1−18=781 - \frac{1}{8} = \frac{7}{8}

Step 3: Q retains his original share

Q's new share = Q's old share = 13\frac{1}{3}

This is the critical constraint. Q does not sacrifice anything.

Step 4: Calculate the combined new share of P and R

Since Q takes 13\frac{1}{3} out of the 78\frac{7}{8} available to the old partners:

Combined share of P and R = 78−13\frac{7}{8} - \frac{1}{3}

To subtract, find a common denominator (24):

78−13=2124−824=1324\frac{7}{8} - \frac{1}{3} = \frac{21}{24} - \frac{8}{24} = \frac{13}{24}

Step 5: P and R will share this 1324\frac{13}{24} in their old ratio of 6:46 : 4 (i.e., 3:23 : 2)

P's new share = 35×1324=39120=1340\frac{3}{5} \times \frac{13}{24} = \frac{39}{120} = \frac{13}{40}

R's new share = 25×1324=26120=1360\frac{2}{5} \times \frac{13}{24} = \frac{26}{120} = \frac{13}{60}

Note

When one partner retains his original share, the remaining partners continue to share the balance in their old mutual ratio.

Working Notes

W.N. 1: Sacrifice by P

P’s sacrifice=Old share−New share=25−1340\text{P's sacrifice} = \text{Old share} - \text{New share} = \frac{2}{5} - \frac{13}{40}

Convert 25\frac{2}{5} to denominator 40: 25=1640\frac{2}{5} = \frac{16}{40}

P’s sacrifice=1640−1340=340\text{P's sacrifice} = \frac{16}{40} - \frac{13}{40} = \frac{3}{40}

W.N. 2: Sacrifice by R

R’s sacrifice=415−1360\text{R's sacrifice} = \frac{4}{15} - \frac{13}{60}

Convert 415\frac{4}{15} to denominator 60: 415=1660\frac{4}{15} = \frac{16}{60}

R’s sacrifice=1660−1360=360=120\text{R's sacrifice} = \frac{16}{60} - \frac{13}{60} = \frac{3}{60} = \frac{1}{20}

W.N. 3: Sacrificing Ratio

Sacrificing Ratio (P : R)=340:120\text{Sacrificing Ratio (P : R)} = \frac{3}{40} : \frac{1}{20}

To compare, convert to a common denominator (40):

340:240=3:2\frac{3}{40} : \frac{2}{40} = 3 : 2

Watch out

A common mistake is to assume the sacrificing ratio equals the old profit-sharing ratio. Here, P and R's old ratio was 6:46 : 4 (or 3:23 : 2), which happens to match the sacrificing ratio — but this is coincidental because Q retained his full share. Always compute sacrifice = old − new for each partner.

✓Final answer

P sacrifices 340\frac{3}{40} and R sacrifices 120\frac{1}{20} (i.e., 240\frac{2}{40}), giving a sacrificing ratio of 3 : 2. The correct answer is (C) 3 : 2.

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