New Profit Sharing Ratio

C receives 1/6 of the profits, so 5/6 remains for A and B. This remainder is divided in the old ratio 7:5. Therefore, A's share is (5/6) × (7/12) = 35/72, and B's share is (5/6) × (5/12) = 25/72. C's share is 1/6 = 12/72. Hence, the new profit-sharing ratio is 35:25:12. Exam tip: When the incoming partner receives a specified share and the balance is divided in the old ratio, multiply each old partner's fraction by the remaining share.

A and B’s old shares are 5/8 and 3/8 respectively. C’s 1/4 share is sacrificed by A and B in the ratio 1:2, so A sacrifices 1/12 and B sacrifices 1/6. Therefore, A’s new share is 5/8 − 1/12 = 13/24, B’s new share is 3/8 − 1/6 = 5/24, and C’s share is 1/4 = 6/24. Hence, the new ratio is 13:5:6. Exam tip: Subtract each existing partner’s sacrifice from the old share and then include the incoming partner’s share.

Initially, A’s share is \(4/9\) and B’s share is \(5/9\). Since C receives \(2/9\) of the total profit from B, A’s share remains \(4/9\), while B’s new share becomes \(5/9-2/9=3/9\). C’s share is \(2/9\). Therefore, the new ratio is \(4/9:3/9:2/9=4:3:2\). Option B is incorrect because it does not deduct C’s share from B’s original share. Exam tip: If the incoming partner acquires a share from only one old partner, reduce the share of only that partner.

Direct answer: Option A, 2:1, is correct. A and B originally share in 3:2, so their total is 5 parts: A’s old share is 3/5 = 9/15 and B’s old share is 2/5 = 6/15. After admission, the new ratio is 7:5:3, whose total is 15. Thus A’s new share is 7/15 and B’s new share is 5/15. A sacrifices 9/15 − 7/15 = 2/15. B sacrifices 6/15 − 5/15 = 1/15. The sacrificing ratio is therefore 2/15:1/15 = 2:1. Option A works. Option B repeats the old profit-sharing ratio, but sacrifice is based on the difference between old and new shares. Option C uses the new shares rather than the amounts surrendered. Option D reverses the calculated sacrifice amounts. Check also that C receives 3/15 and the final shares total 15/15. Memory cue: put old and new shares over a common denominator, subtract, then simplify the sacrifices.

The old ratio 11:5 has a total of 16 parts, so A’s old share is 11/16 and B’s share is 5/16. C receives 1/4 = 4/16 exclusively from A; therefore, A’s new share is 11/16 − 4/16 = 7/16. B’s share remains 5/16, while C receives 4/16. Hence, the new ratio is 7:5:4. Exam tip: When the incoming partner obtains the entire share from only one old partner, only that partner’s share is reduced.

C receives 1/3 of the total profit, so the remaining share for A and B is 2/3. A’s new share is \(\frac{2}{3}\times\frac{2}{9}=\frac{4}{27}\), while B’s new share is \(\frac{2}{3}\times\frac{7}{9}=\frac{14}{27}\). C’s share is \(\frac{1}{3}=\frac{9}{27}\). Therefore, the new ratio is 4:14:9. Exam tip: First subtract the incoming partner’s share from 1, then distribute the balance among the old partners in their old ratio.

The governing concept is comparison of old and new shares after admission. Before C entered, A and B shared the total in the ratio 5:4, so A’s share was 5/9 and B’s was 4/9. The new ratio 4:3:2 totals 9, giving A 4/9, B 3/9, and C 2/9. A’s sacrifice is therefore 5/9 − 4/9 = 1/9. B’s sacrifice is 4/9 − 3/9 = 1/9. The combined sacrifice is 2/9, exactly equal to C’s new share. Since both A and B surrender the same amount, C’s share has come through equal sacrifice. Hence option A is correct; options B and C ignore one of the equal reductions, while D does not describe the calculated source.

B’s old share is \(\frac{5}{11}\). Since the total sacrifice of \(\frac{2}{11}\) is divided between A and B in the ratio 3:1, B’s sacrifice is \(\frac{2}{11}\times\frac{1}{3+1}=\frac{1}{22}\). Therefore, B’s new share is \(\frac{5}{11}-\frac{1}{22}=\frac{9}{22}\). Exam tip: A continuing partner’s new share equals the old share minus the share sacrificed.

A’s old share was 7/10, which decreased to 5/10, so A sacrificed 2/10. B’s old share was 3/10 and its new share is also 3/10; therefore, B made no sacrifice. The new partner C gained 2/10. Exam tip: calculate sacrifice by subtracting the new share from the old share.

A’s old share is 3/8 and B’s old share is 5/8. C’s 1/4 share is contributed by A and B in the sacrificing ratio 1:3; therefore, A sacrifices 1/16 and B sacrifices 3/16. Thus, A’s new share is 3/8 − 1/16 = 5/16, B’s new share is 5/8 − 3/16 = 7/16, and C’s share is 1/4 = 4/16. Hence, the new profit-sharing ratio is 5:7:4. Exam tip: Convert the old ratio into fractions first, then distribute the incoming partner’s share according to the sacrificing ratio and subtract it. Option B fails to make this proper sacrifice adjustment.

Under the old ratio 4:7, A’s share is \(\frac{4}{11}\) and B’s share is \(\frac{7}{11}\). Since C receives \(\frac{2}{11}\) entirely from B, B’s new share becomes \(\frac{7}{11}-\frac{2}{11}=\frac{5}{11}\). Thus, the new shares of A, B and C are \(\frac{4}{11}:\frac{5}{11}:\frac{2}{11}\), giving the new ratio 4:5:2. Option B is incorrect because it does not deduct C’s share from B. Exam tip: If an incoming partner obtains a share from only one old partner, subtract that share from that partner’s old share and leave the others unchanged.

C receives 1/19 of the total profit, so the remaining profit for A and B is 18/19. This remainder is divided in the old ratio 10:9. Therefore, A receives (18/19) × (10/19) = 180/361 and B receives (18/19) × (9/19) = 162/361. C’s share, 1/19, is equal to 19/361. Hence, the new ratio is 180:162:19. Option B incorrectly combines the old ratio directly with C’s share without reducing A’s and B’s shares. Exam tip: If the incoming partner receives a specified share and the balance is divided in the old ratio, multiply each old partner’s fractional share by the remaining profit fraction.

C’s share is \(\frac{1}{7}=\frac{7}{49}\). In the sacrificing ratio 2:5, A sacrifices \(\frac{2}{49}\) and B sacrifices \(\frac{5}{49}\). Their old shares were \(\frac{5}{7}=\frac{35}{49}\) and \(\frac{2}{7}=\frac{14}{49}\), respectively. Thus, A’s new share is \(\frac{33}{49}\), B’s is \(\frac{9}{49}\), and C’s is \(\frac{7}{49}\), giving the ratio 33:9:7. Exam tip: Distribute the incoming partner’s share in the sacrificing ratio and subtract each sacrifice from the respective old share.

The total of the new ratio 13:11:6 is 30. Therefore, C's new share is 6/30 = 1/5. The old ratio is not required for this calculation because C's share is given directly in the new ratio. Exam tip: To find a partner's share under a new ratio, divide that partner's ratio term by the sum of all the new-ratio terms.

The partners’ new shares must add up to 1. Therefore, B’s share is \(1-\frac{7}{15}-\frac{1}{5}=1-\frac{7}{15}-\frac{3}{15}=\frac{5}{15}=\frac{1}{3}\).

Sacrifice is the portion of profit given up by an old partner; hence it equals old share minus new share, \(O_i-N_i\). For example, \(3/10-1/5=1/10\). In exams, first express old and new shares with a common denominator.

A’s old share is \(\frac{3}{3+4}=\frac{3}{7}\). A bears \(\frac{2}{5}\) of C’s 2/7 share because the sacrifice ratio is 2:3, so A’s sacrifice is \(\frac{2}{5}\times\frac{2}{7}=\frac{4}{35}\). A’s new share is therefore \(\frac{3}{7}-\frac{4}{35}=\frac{11}{35}\).

The old ratio 8:5 gives A a share of 8/13 and B a share of 5/13. C’s 3/13 share is sacrificed by A and B in the ratio 2:1, so A sacrifices 2/13 and B sacrifices 1/13. Therefore, A’s new share is 8/13 − 2/13 = 6/13, B’s is 5/13 − 1/13 = 4/13, and C’s is 3/13. Hence, the new ratio is 6:4:3. Exam tip: subtract each partner’s sacrifice from their old share; do not simply append the newcomer’s share to the old ratio.

C receives 3/13 of the profit, so the remaining share for A and B is 10/13. Distributing this remainder in the old ratio 3:10 gives A: (10/13) × (3/13) = 30/169 and B: (10/13) × (10/13) = 100/169. C’s share, 3/13, equals 39/169. Therefore, the new ratio is 30:100:39. Exam tip: Multiply each old-ratio term by the remaining fraction, then express all three shares with a common denominator.

Under the old ratio, A’s share was \(\frac{7}{9}\) and B’s share was \(\frac{2}{9}\). Under the new ratio 5:2:2, A gets \(\frac{5}{9}\), B remains at \(\frac{2}{9}\), and C gets \(\frac{2}{9}\). A therefore sacrifices \(\frac{7}{9}-\frac{5}{9}=\frac{2}{9}\), while B sacrifices nothing. Hence, C’s entire share comes from A; option B is incorrect because B’s share does not change. Exam tip: Compare each old partner’s old and new fractions to identify who has sacrificed a share.

The total number of ratio parts is 11+9+5=25. G's share is 5/25=1/5. The combined share of old partners A and B is (11+9)/25=20/25=4/5. Option B incorrectly gives their combined share as 15/25; the correct sum is 20/25. Exam tip: Add the ratio parts of the relevant old partners first, then divide by the total parts.

The old ratio 4:11 gives A and B shares of 4/15 and 11/15, respectively. Since C receives 2/15 entirely from B, A’s share remains 4/15, while B’s new share becomes 11/15 − 2/15 = 9/15. Thus, the new ratio of A, B and C is 4:9:2. Exam tip: When the incoming partner obtains a share from only one existing partner, only that partner’s share is reduced.

Since the old ratio is 5:11, A’s and B’s old shares are 5/16 and 11/16. C receives 1/4, so A sacrifices 1/20 and B sacrifices 1/5 because the sacrificing ratio is 1:4. Therefore, A’s new share is 5/16 − 1/20 = 21/80, B’s new share is 11/16 − 1/5 = 39/80, and C’s share is 1/4 = 20/80. Hence, the new ratio is 21:39:20. Exam tip: Subtract each old partner’s sacrifice and add the admitted partner’s share; verify that the resulting shares total 1.

The old ratio 2:9 has a total of 11 parts, so B’s old share is \(\frac{9}{11}\). The total share sacrificed for C is \(\frac{3}{11}\). Since A:B sacrifice in the ratio 1:5, B’s sacrifice is \(\frac{3}{11}\times\frac{5}{6}=\frac{5}{22}\). Therefore, B’s new share is \(\frac{9}{11}-\frac{5}{22}=\frac{18}{22}-\frac{5}{22}=\frac{13}{22}\). Exam tip: A partner’s new share equals the old share minus the share sacrificed for the incoming partner.

The governing concept is proportional distribution of the residual profit. C receives 2/11, so the profit remaining for A and B together is 1 − 2/11 = 9/11. This remainder is divided in their old ratio 8:3, whose total is 11. Therefore A receives (9/11) × (8/11) = 72/121, while B receives (9/11) × (3/11) = 27/121. C’s share, 2/11, expressed with denominator 121, is 22/121. The common-denominator ratio is therefore 72:27:22, making option A correct. The shares add to 121/121, confirming the calculation. Options B, C and D either use the old ratio without adjusting the remainder or fail to distribute 9/11 proportionately.