New Profit Sharing Ratio

A and B's old profit shares are \(\frac{5}{12}\) and \(\frac{7}{12}\), respectively. Out of C's \(\frac{1}{4}\) share, B sacrifices \(\frac{1}{4}\times\frac{2}{3}=\frac{1}{6}\), because the sacrificing ratio is 1:2. Therefore, B's new share is \(\frac{7}{12}-\frac{1}{6}=\frac{5}{12}\). Exam tip: subtract each partner's sacrifice from that partner's old share; do not subtract the entire incoming partner's share.

C receives 1/6 of the total profit, leaving 5/6 for A and B. They divide this remainder in their old ratio of 11:7, so A receives (5/6 × 11/18) = 55/108 and B receives (5/6 × 7/18) = 35/108. C's share, 1/6, equals 18/108. Therefore, the new ratio A:B:C is 55:35:18.

C receives 3/13, so the remaining share for A and B is 10/13. Distributing this in the old ratio 4:9 gives A = 10/13 × 4/13 = 40/169 and B = 10/13 × 9/13 = 90/169. C’s share, 3/13, equals 39/169. Therefore, the new ratio is 40:90:39. Exam tip: First calculate the remaining profit, then distribute it among the old partners in their old ratio.

C receives 2/11 of the total profit, so the remaining share for A and B is 1 − 2/11 = 9/11. This balance is divided in the old ratio 8:3. Therefore, A’s share is (9/11) × (8/11) = 72/121, and B’s share is (9/11) × (3/11) = 27/121. C’s share, 2/11, is 22/121. Hence, the new profit-sharing ratio is 72:27:22. Option A is incorrect because C’s share cannot simply be appended to the old ratio. Exam tip: First subtract the admitted partner’s share from 1, then distribute the balance among the old partners in their old ratio.

C receives \(\frac{1}{4}\) of the total profit, so the remaining profit for A and B is \(\frac{3}{4}\). The sum of A and B’s old ratio is 11. Therefore, A’s share is \(\frac{3}{4}\times\frac{5}{11}=\frac{15}{44}\), and B’s share is \(\frac{3}{4}\times\frac{6}{11}=\frac{18}{44}\). C’s share is \(\frac{1}{4}=\frac{11}{44}\). Hence, the new ratio is \(15:18:11\). Exam tip: First calculate the balance share, then distribute it among the old partners in their old ratio.

C receives 3/11, leaving 8/11 for A and B. They divide this remainder in their old ratio of 7:4, so A receives (8/11 × 7/11) = 56/121 and B receives (8/11 × 4/11) = 32/121. C’s share, 3/11, equals 33/121. Therefore, the new ratio is 56:32:33.

C receives 2/9, so the profit remaining for A and B is 1 − 2/9 = 7/9. This balance is divided in the old ratio 2:7: A receives (7/9) × (2/9) = 14/81 and B receives (7/9) × (7/9) = 49/81. C’s share, 2/9, equals 18/81. Therefore, the new ratio is 14:49:18. Exam tip: When the incoming partner receives a specified share and the balance remains with the old partners, multiply each old ratio fraction by the remaining share.

C receives \(1/11\), so the profit left for A and B is \(10/11\). Distributing this remainder in the old ratio \(9:2\), A receives \(10/11 \times 9/11=90/121\) and B receives \(10/11 \times 2/11=20/121\). C’s share is \(1/11=11/121\). Therefore, the new ratio is \(90:20:11\). Option A incorrectly treats the old ratio as unchanged after C’s admission. Exam tip: subtract the new partner’s share from 1, then multiply the remainder by each old partner’s share.


C receives 3/13, so the remaining profit is 1 − 3/13 = 10/13. This balance is divided between A and B in their old ratio of 3:10. Therefore, A receives (10/13) × (3/13) = 30/169 and B receives (10/13) × (10/13) = 100/169. C’s share, 3/13, equals 39/169. Hence, the new ratio is 30:100:39. The key exam tip is to calculate the balance profit first and then apply the old ratio to it.

C receives \(1/9\) of the total profit, so the remaining share for A and B is \(8/9\). The sum of the old ratio 13:5 is 18. Therefore, A’s share is \((8/9)\times(13/18)=52/81\), and B’s share is \((8/9)\times(5/18)=20/81\). C’s share is \(1/9=9/81\). Hence, the new ratio is \(52:20:9\). For instance, in 104:40:9, C’s share would be \(9/153=1/17\), not \(1/9\), so that option is incorrect. Exam tip: subtract the incoming partner’s share first, then distribute the balance in the old ratio.

C’s share is 4/13, so the remaining share for A and B is 1 − 4/13 = 9/13. Distributing this remainder in the old ratio 6:7 gives A’s share as (9/13) × (6/13) = 54/169 and B’s share as (9/13) × (7/13) = 63/169. C’s share, 4/13, equals 52/169. Therefore, the new ratio is 54:63:52. Exam tip: When the incoming partner’s share is given and the remainder is divided in the old ratio, multiply the remainder by each old partner’s fractional share.

The old ratio of A and B is 5:8, so A’s original share is \(5/13\) and B’s share is \(8/13\). Since C receives \(2/13\) entirely from A, A’s new share becomes \(5/13-2/13=3/13\). B’s share remains \(8/13\), while C receives \(2/13\). Therefore, the new profit-sharing ratio is \(3:8:2\). Exam tip: When an incoming partner obtains the entire share from one old partner, only that partner’s share is reduced.

When the new partner receives the full share from only one partner, that partner alone sacrifices. In the new ratio, subtract the new partner’s share from that partner’s old share; the other partner’s share is unchanged. Exam tip: identify the sacrificing partner first.

The old ratio of A and B is 7:6, so their original shares are \(7/13\) and \(6/13\), respectively. C receives \(1/13\) only from A; therefore, A’s new share is \(7/13-1/13=6/13\). B’s share remains \(6/13\), while C receives \(1/13\). Hence, the new profit-sharing ratio is \(6:6:1\). Exam tip: When the incoming partner acquires the entire share from one partner, subtract that share only from the identified partner’s original share.

The total of the old ratio 4:11 is 15, so A's share is 4/15 and B's share is 11/15. Since C acquires 2/15 entirely from B, B's new share becomes 11/15 − 2/15 = 9/15. Therefore, the new ratio A:B:C is 4/15:9/15:2/15 = 4:9:2. Exam tip: When the incoming partner acquires the entire share from only one old partner, the other old partner's share remains unchanged.

The old shares of A and B are 10/17 and 7/17 respectively. Since C receives 3/17 exclusively from A, A’s new share becomes 10/17 − 3/17 = 7/17, while B’s share remains 7/17. Therefore, the new ratio is 7/17 : 7/17 : 3/17 = 7:7:3. Exam tip: When the incoming partner obtains the share from only one old partner, subtract the sacrificed share only from that partner’s share.

If the new partner receives the entire share from one partner, only that partner sacrifices profit. The partner making no sacrifice keeps the same share. In exams, compare old and new shares to identify the sacrificing partner.

The old ratio 12:5 has a total of 17 parts, so A's share is 12/17 and B's share is 5/17. Since C receives 2/17 entirely from A, A's new share becomes 12/17 − 2/17 = 10/17. B's share remains 5/17, while C gets 2/17. Therefore, the new ratio is 10:5:2. Exam tip: If the new partner receives a share from only one existing partner, reduce the share of only that partner.

The old total ratio is \(3+14=17\), so A’s share is \(3/17\) and B’s share is \(14/17\). Since C’s \(1/17\) share is taken entirely from B, A’s share remains unchanged, while B’s new share becomes \(14/17-1/17=13/17\). Therefore, the new ratio is \(3/17:13/17:1/17=3:13:1\). Exam tip: If the incoming partner takes the entire share from only one existing partner, reduce the share of only that partner.

C's (1/3) share is taken in (1:2). A sacrifices (1/9) and B sacrifices (2/9) so the ratio becomes (4:2:3).

The old ratio 7:5 gives A a share of 7/12 and B a share of 5/12. C’s 1/4 share is surrendered by A and B in the ratio 3:1, so A sacrifices 3/16 and B sacrifices 1/16. Therefore, A’s new share is 7/12 − 3/16 = 19/48, B’s is 5/12 − 1/16 = 17/48, and C’s is 1/4 = 12/48. Hence, the new ratio is 19:17:12. Exam tip: subtract each partner’s sacrifice from the old share and express all resulting shares with a common denominator.

A’s old share is 4/11 and B’s old share is 7/11. C’s 2/11 share is sacrificed in the ratio 1:3, so A sacrifices 1/22 and B sacrifices 3/22. Therefore, A’s new share is 8/22−1/22=7/22, B’s is 14/22−3/22=11/22, and C’s share is 4/22. Hence, the new ratio is 7:11:4. Exam tip: Subtract each existing partner’s sacrifice from the old share and assign the incoming partner’s share directly.

A and B previously held \(\frac{8}{13}\) and \(\frac{5}{13}\), respectively. C’s \(\frac{3}{13}\) share is sacrificed by A and B in the ratio 2:1, so A sacrifices \(\frac{2}{13}\) and B sacrifices \(\frac{1}{13}\). Their new shares are therefore \(\frac{6}{13}\) and \(\frac{4}{13}\), while C receives \(\frac{3}{13}\). Hence, the new ratio is 6:4:3. Exam tip: subtract each partner’s sacrificed share from the old share, then include the incoming partner’s share.

C’s 2/13 share is sacrificed by A and B in the ratio 5:3. Therefore, A’s sacrifice is 2/13 × 5/8 = 5/52, and B’s sacrifice is 2/13 × 3/8 = 3/52. Converting the old ratio 6:7 to denominator 52 gives A’s and B’s old shares as 24/52 and 28/52. Their new shares are 19/52 and 25/52, while C receives 8/52. Hence, the new ratio is 19:25:8. Exam tip: Deduct the incoming partner’s share from the old partners in the given sacrificing ratio.

Since the old ratio is 9:5, A and B had shares \(\frac{9}{14}\) and \(\frac{5}{14}\). C receives \(\frac{1}{7}\), and this is sacrificed in the ratio 4:3; therefore, A sacrifices \(\frac{4}{49}\) and B sacrifices \(\frac{3}{49}\). Their new shares are \(\frac{9}{14}-\frac{4}{49}=\frac{55}{98}\) and \(\frac{5}{14}-\frac{3}{49}=\frac{29}{98}\), while C receives \(\frac{1}{7}=\frac{14}{98}\). Hence, the new profit-sharing ratio is 55:29:14. Exam tip: subtract each existing partner’s sacrificed share from the old share; do not subtract the new partner’s share from the old ratio directly.