New Profit Sharing Ratio

A new partner’s share is obtained from the sacrifice made by the old partners. Hence, an old partner’s new share equals old share minus sacrificing share. In exams, find the sacrificing share first and subtract it from the old share.

The direct answer is 9:5:2, so option A is correct. C’s share is 1/8, and it is taken equally from A and B. Therefore each old partner sacrifices half of 1/8, or 1/16. Convert the old shares to sixteenths: A’s 5/8 equals 10/16, and B’s 3/8 equals 6/16. After sacrifice, A has 10/16−1/16=9/16, while B has 6/16−1/16=5/16. C has 1/8=2/16. Thus the new ratio is 9:5:2. The total 9+5+2=16 confirms that the shares represent the whole profit. Option B does not deduct the sacrifice. Option C gives the old partners an incorrect reduction and does not represent C’s one-eighth share properly. Option D assigns C 1/16 instead of 2/16. The common confusion is to subtract 1/8 from each partner; that would transfer 1/4 in total, twice the amount C should receive. Equal taking means each gives 1/16.

The old ratio of A and B is 7:1, so the whole profit is divided into eight equal parts. C receives one-fourth of the total profit. This new share is taken equally from A and B, so each old partner sacrifices one-eighth. A originally had seven-eighths and gives up one-eighth, leaving six-eighths. B originally had one-eighth and gives up one-eighth, leaving no share. C receives two-eighths, because one-fourth equals two-eighths.

Therefore, the new shares are A = 6/8, B = 0/8 and C = 2/8. Written in ratio form, the new ratio is 6:0:2, which is option A. It may look unusual that B’s share becomes zero, but it follows directly from B’s old share and his equal sacrifice. The ratio could be simplified mathematically to 3:0:1, but the supplied option correctly preserves the common denominator.

Take the total profit as 1. A’s and C’s new shares are \(\frac{7}{15}\) and \(\frac{1}{5}=\frac{3}{15}\), respectively. Therefore, B’s share is \(1-\frac{7}{15}-\frac{3}{15}=\frac{5}{15}=\frac{1}{3}\). The old ratio \(3:2\) is not used directly because A’s and C’s new shares are already given. Exam tip: Always check that the sum of all partners’ new shares equals 1.

The sum of all partners’ new profit shares must be 1. Therefore, B’s new share = 1 − 3/7 − 2/7 = 2/7. Hence, option A is correct. In such questions, always verify that the new profit shares add up to 1.

The sum of the new profit-sharing ratio 3:6:3 is 12. C’s corresponding ratio part is 3, so C’s share is 3/12 = 1/4. The option 1/3 is incorrect because it results from using an incorrect total of 9 instead of 12. Exam tip: Divide the concerned partner’s ratio part by the sum of all the ratio parts.

The ratio (2:2:1) shows A and B share the balance equally. In exams convert the ratio into shares to check.

The new ratio depends on from whom the new partner gets the share. So first identify the share and source of sacrifice.

Sacrifice in old ratio means the balance profit is also shared in the old ratio. This is a very common exam pattern.

C receives 1/3 of the total profit, leaving 2/3 for A and B. Sharing this balance in their old ratio of 1:2 gives A \(\frac{2}{3} \times \frac{1}{3}=\frac{2}{9}\) and B \(\frac{2}{3} \times \frac{2}{3}=\frac{4}{9}\). C’s share is \(\frac{1}{3}=\frac{3}{9}\). Therefore, the new ratio A:B:C is 2:4:3. Exam tip: If the incoming partner takes a specified share and the balance is divided in the old ratio, multiply each old partner’s fraction by the remaining share.

C receives 3/17 of the profit, leaving 14/17 for A and B. Since they divide this remainder in their old ratio of 10:7, A’s share is (14/17) × (10/17) = 140/289, and B’s is (14/17) × (7/17) = 98/289. C’s share is 3/17 = 51/289. Therefore, the new profit-sharing ratio is 140:98:51.

The old ratio 3:7 means A’s share is 3/10 and B’s share is 7/10. Since C’s 1/10 share is taken entirely from B, A’s share remains 3/10, while B’s new share becomes 7/10 − 1/10 = 6/10. Therefore, A:B:C = 3/10:6/10:1/10 = 3:6:1. Exam tip: when the new partner’s share is taken from only one existing partner, reduce only that partner’s share.

A's old share was (4/5). After sacrificing (1/5), A's new share is (3/5) so the ratio is (3:1:1).

A and B originally have shares of 4/10 and 6/10, which are 8/20 and 12/20. C receives 1/5, or 4/20. Since the sacrificing ratio is 1:3, A sacrifices 1/20 and B sacrifices 3/20. Thus, A’s new share is 7/20, B’s is 9/20, and C’s is 4/20, giving the new ratio 7:9:4. Exam tip: subtract each old partner’s sacrifice from the old share and assign the admitted partner’s share accordingly.

A sacrifices (2/15) and B sacrifices (1/30). New shares are (7/10,2/15,1/6), which become (21:4:5).

The direct answer is 9:17:6, so option A is correct. C receives 3/16, taken equally from A and B, so each old partner sacrifices 3/32. Express the old shares with denominator 32: A’s 3/8 becomes 12/32, and B’s 5/8 becomes 20/32. After sacrifice, A receives 12/32−3/32=9/32 and B receives 20/32−3/32=17/32. C receives 3/16=6/32. Hence the new ratio is 9:17:6. The check is 9/32+17/32+6/32=32/32=1, so no profit is missing or counted twice. Option B fails to account for the equal sacrifice and gives C the wrong effective part. Option C reduces the old shares incorrectly and gives an incorrect C component. Option D leaves the old ratio merely scaled and does not subtract the amount transferred to C. A useful warning is that half of 3/16 is 3/32, not 3/16; forgetting this is the main source of error in equal-sacrifice questions.

C receives \(2/13\), so the remaining share for A and B is \(1-2/13=11/13\). Sharing this remainder in the old ratio 6:7 gives A \((11/13)×(6/13)=66/169\) and B \((11/13)×(7/13)=77/169\). C’s share is \(2/13=26/169\). Therefore, the new ratio is 66:77:26. Exam tip: first subtract the incoming partner’s share from 1, then distribute the remainder in the old ratio.

The total ratio is \(5+3+2=10\). C’s ratio share is 2, so C’s share of the profit is \(\frac{2}{10}=\frac{1}{5}\). Option B incorrectly uses 5 as the denominator instead of the total ratio. Exam tip: Divide a partner’s ratio by the sum of all partners’ ratio terms to find that partner’s profit share.

The total parts in the new ratio 7:5:3 are \(7+5+3=15\). The old partners A and B have 7 and 5 parts respectively, so their combined share is \(7+5=12\) parts. Therefore, their combined profit share is \(\frac{12}{15}=\frac{4}{5}\). Exam tip: Add the ratio terms of the specified partners and divide by the total of all ratio terms.

The sum of the new ratio 3:4:2 is 3 + 4 + 2 = 9. C receives 2 parts, so C’s share in the total profit is 2/9. The value 1/9 represents only one part, while 2/7 results from using an incorrect total. Exam tip: add all terms of the new ratio and divide the concerned partner’s term by that total.

The sum of the terms in the new profit-sharing ratio 4:3:3 is 10. C is represented by the third term, 3. Therefore, C’s share is \(\frac{3}{4+3+3}=\frac{3}{10}\). \(\frac{1}{3}\) is incorrect because the total of the ratio terms is 10, not 9. Exam tip: A partner’s share equals that partner’s ratio term divided by the sum of all ratio terms.

The sum of the terms in the new profit-sharing ratio 7:6:5 is 18. Since A’s term is 7, A’s new share is \(\frac{7}{18}\). The old ratio 5:4 is not needed because the complete new ratio is already given. Exam tip: A partner’s share equals that partner’s ratio term divided by the sum of all new ratio terms.

B’s old share was 3/5 (from the old ratio 2:3), and B’s new share is 2/5 (from the new ratio 2:2:1). B’s sacrifice is the old share minus the new share: 3/5 − 2/5 = 1/5.

A’s old profit share was \(\frac{4}{5}\), since the old ratio was 4:1. In the new ratio 3:1:1, A’s new share is \(\frac{3}{5}\). Therefore, A’s sacrifice = old share − new share = \(\frac{4}{5}-\frac{3}{5}=\frac{1}{5}\). Option D, \(\frac{3}{5}\), is A’s new share, not the sacrifice. Exam tip: Sacrifice is always calculated as old share minus new share.

Under the old ratio of 3:1, B’s share was 1/4. In the new ratio of 2:1:1, B’s share remains 1/4. Therefore, sacrifice = old share − new share = 1/4 − 1/4 = zero. Hence, B made no sacrifice; C’s 1/4 share is effectively adjusted from A’s reduced share. Exam tip: Calculate each partner’s sacrifice separately as old share minus new share.