New Profit Sharing Ratio

B's old share was 1/5. After C receives 2/5, 3/5 remains for A and B. B gets 2/3 of that remainder, so B's new share is (2/3 × 3/5) = 2/5. B's gain is the new share minus the old share: 2/5 − 1/5 = 1/5.

The incoming partner’s share is equal to the total sacrifice made by the old partners. Therefore, A and B together sacrifice 1/4 = 4/16. Since A’s sacrifice is 1/16, B’s sacrifice is 4/16 − 1/16 = 3/16. Option 1/8 is incorrect because the total sacrifice need not be divided equally between the old partners. Exam tip: First find the total sacrifice equal to the incoming partner’s share, then subtract the given partner’s sacrifice.

C's \(\frac{1}{4}\) share is sacrificed by A and B in the ratio 1 : 3. Thus, A sacrifices \(\frac{1}{16}\) and B sacrifices \(\frac{3}{16}\). A's old share was \(\frac{5}{8}=\frac{10}{16}\), so the new share is \(\frac{9}{16}\). B's new share is \(\frac{3}{8}-\frac{3}{16}=\frac{3}{16}\), while C receives \(\frac{4}{16}\). Hence, the new ratio is 9 : 3 : 4. Option B fails to deduct the correct sacrifice from the old shares. Exam tip: The total reduction in the old partners' shares must equal the new partner's admitted share.

D's share comes from A and C in (3:1) so A sacrifices (3/20) and C (1/20). New shares are (7/20) (6/20) (3/20) and (4/20).

The direct answer is A, 5/18. A and B originally share 5:4, so B’s old share is 4/9. C receives 1/3, leaving 2/3 for the old partners. Their new mutual ratio is 3:1, so B receives one of four parts of the remaining 2/3: (2/3) × (1/4) = 1/6. B’s sacrifice is old share minus new share: 4/9 − 1/6 = 8/18 − 3/18 = 5/18. Option A is correct. Option B, 1/9, is too small and does not result from subtraction. Option C, 1/3, is C’s admitted share, not B’s sacrifice. Option D, 4/9, is B’s old share, not the amount surrendered. Exam cue: first distribute the remaining share according to the new mutual ratio.

The governing rule is that sacrifice equals the old share minus the new share. A’s old share under the 5:4 ratio was 5/9. After C receives 1/3, the balance for A and B is 2/3. Their new mutual ratio is 3:1, so A receives 3/4 of that balance: (2/3) × (3/4) = 1/2. Therefore A’s sacrifice is 5/9 − 1/2. Taking 18 as the common denominator gives 10/18 − 9/18 = 1/18. Hence option A is correct. Option B is not the actual difference, option C is C’s share, and option D is A’s old share. The result also confirms that A’s share decreases after admission.

C receives \(1/3\) of the profit, so the balance left for A and B is \(1 - 1/3 = 2/3\). This balance is divided in the ratio 3:1; therefore, A receives \(2/3 \times 3/4 = 1/2\) and B receives \(2/3 \times 1/4 = 1/6\). Thus, A:B:C = \(1/2:1/6:1/3\) = 3:1:2. Option B incorrectly uses the old ratio 7:2 directly as part of the new ratio. Exam tip: subtract the incoming partner’s share from one, then divide the remaining share according to the specified ratio of the old partners.

A falls from (7/9) to (1/2) and B falls from (2/9) to (1/6). Both shares decrease so no old partner gained.

The governing principle is that when the incoming partner’s entire share is taken from one old partner, only that partner’s share changes. A’s old share under the ratio 8:1 was 8/9. C receives 2/9 entirely from A, so A’s new share becomes 8/9 − 2/9 = 6/9. B does not sacrifice and retains his old share of 1/9. The new mutual ratio of A and B is therefore 6/9:1/9 = 6:1. Hence option C is correct. Option B is merely the old ratio, option D does not subtract the full amount transferred by A, and option A results from an incorrect reduction. The shares also remain consistent with C’s 2/9 admission share.

In the old ratio 8:1, A’s share was 8/9 and B’s share was 1/9. Since C’s 2/9 share is taken entirely from A, A’s new share becomes 8/9 − 2/9 = 6/9, while B’s share remains 1/9. Therefore, A and B’s new mutual ratio is 6/9 : 1/9 = 6:1. Exam tip: when the incoming partner acquires the share entirely from one existing partner, reduce only that partner’s share.

B's sacrifice is 1/5 × 1/4 = 1/20, because B contributes one-fourth of the total sacrifice under the 3:1 sacrifice ratio. B's old share was 8/25. Therefore, B's new share = 8/25 − 1/20 = 32/100 − 5/100 = 27/100. Option 8/25 is the old share and does not account for B's sacrifice. Exam tip: first divide the incoming partner's share according to the sacrifice ratio, then subtract each partner's sacrifice from the old share.

C’s share is \(1/5\), and the sacrificing ratio is 3:1. Therefore, A sacrifices \(3/20\) and B sacrifices \(1/20\). A’s new share is \(17/25-3/20=53/100\), while B’s new share is \(8/25-1/20=27/100\). C’s share is \(1/5=20/100\). Hence, the new ratio is \(53:27:20\). Exam tip: first distribute the incoming partner’s share according to the sacrificing ratio, then subtract each sacrifice from the corresponding old share.

D’s 1/10 share is taken equally from A and C, so each sacrifices 1/20. The original shares are A = 6/20, B = 10/20 and C = 4/20. After the sacrifice, their shares are A = 5/20, B = 10/20 and C = 3/20; D receives 2/20. Therefore, the new profit-sharing ratio is 5:10:3:2.

Sacrificing ratio = A’s sacrifice : B’s sacrifice = 1/12 : 1/12 = 1:1. Hence, option A is correct. The ratio 7:5 is the old profit-sharing ratio, not the sacrificing ratio. In an exam, compare the actual shares sacrificed by the old partners, rather than their old profit-sharing ratio.

A and B’s old shares are \(7/12\) and \(5/12\), respectively. C receives \(1/6=2/12\) of the total profit. Since A and B sacrifice this share equally, each sacrifices \(1/12\). Therefore, A’s new share is \(7/12-1/12=6/12\), B’s is \(5/12-1/12=4/12\), and C’s is \(2/12\). The new ratio is \(6:4:2\), which simplifies to \(3:2:1\). Exam tip: Always verify that the new shares add up to the whole profit. The unchanged ratio in option D ignores the sacrifice made by A and B.

A and B’s old shares were 3/7 and 4/7, respectively. Since C receives 1/7 and A’s new share is 2/5, B’s new share is 1 − 2/5 − 1/7 = 16/35. Therefore, B’s sacrifice = old share − new share = 4/7 − 16/35 = 20/35 − 16/35 = 4/35. Hence, option A is correct. Exam tip: Calculate a partner’s sacrifice by subtracting the new share from the old share.

The old shares are A = 3/7 and B = 4/7. A’s sacrifice is 3/7 − 2/5 = 1/35. Since C receives 1/7 = 5/35, B’s sacrifice is 5/35 − 1/35 = 4/35. Therefore, A’s and B’s sacrificing ratio is 1/35 : 4/35 = 1:4. The ratio 3:4 is only the old profit-sharing ratio, not the sacrificing ratio. Exam tip: Sacrifice of an old partner = old share − new share.

The partners’ new shares must add up to 1. So A’s share is 1 − 1/2 − 2/11 = 22/22 − 11/22 − 4/22 = 7/22. The original 5:6 ratio does not change this calculation because B’s new share is given.

A’s old profit share was \(\frac{5}{11}=\frac{10}{22}\). C receives \(\frac{2}{11}=\frac{4}{22}\), and B’s new share is \(\frac{1}{2}=\frac{11}{22}\). Therefore, A’s new share is \(1-\frac{4}{22}-\frac{11}{22}=\frac{7}{22}\). Hence, A’s sacrifice is \(\frac{10}{22}-\frac{7}{22}=\frac{3}{22}\). The share \(\frac{2}{11}\) belongs to C and is not A’s sacrifice. Exam tip: Sacrifice = old share − new share.

B’s old profit share was \(\frac{6}{11}=\frac{12}{22}\), whereas his new share is \(\frac{1}{2}=\frac{11}{22}\). Since the new share is smaller by \(\frac{1}{22}\), B sacrifices \(\frac{1}{22}\). Use the rule \(\text{Sacrifice} = \text{Old Share} - \text{New Share}\); if the new share is larger, it represents a gain.

With the old ratio 2:3:1, A, B and C hold 6/18, 9/18 and 3/18 respectively. D’s share is 1/6 = 3/18. Since B and C surrender it in the ratio 2:1, B surrenders 2/18 and C surrenders 1/18. Their new shares are therefore 6/18, 7/18, 2/18 and 3/18, giving the new ratio 6:7:2:3. Exam tip: distribute the incoming partner’s share among the sacrificing partners in the stated surrender ratio before subtracting it.

The old ratio of A and B is 15:10 = 3:2. After C receives 1/5, the old partners together retain 4/5. Therefore, A’s new share is 3/5 × 4/5 = 12/25 and B’s new share is 2/5 × 4/5 = 8/25. C’s share is 1/5 = 5/25. Hence, the new ratio is 12:8:5. Option C incorrectly retains the old ratio without reducing A’s and B’s shares for C’s admission. Exam tip: when the new partner’s share is sacrificed by old partners in the old ratio, multiply each old share by (1 − the new partner’s share).

A’s and B’s old shares are 1/6 and 5/6, respectively. Since C receives 1/3 and A and B sacrifice equally, each sacrifices 1/3 ÷ 2 = 1/6. Therefore, B’s new share is 5/6 − 1/6 = 4/6 = 2/3. Hence, option C is correct. Exam tip: When sacrifice is equal, divide the incoming partner’s share equally among the sacrificing partners before subtracting it from each old share.

Answer: option A, 2/3. The incoming partner C receives 1/3 of the total profit. Since A and B sacrifice equally, the total sacrifice of 1/3 is divided equally between them; each sacrifices 1/6. Before admission, B’s share from the ratio 1:5 was 5/(1+5)=5/6. Subtracting B’s sacrifice gives B’s new share=5/6−1/6=4/6=2/3. Therefore A is correct. Option C, 5/6, is B’s old share before C entered. Option D, 1/3, is C’s share. Option B, 1/2, does not follow from either the old ratio or the equal-sacrifice condition. A check confirms the result: A’s old share was 1/6 and he sacrifices 1/6, so A receives zero; B receives 2/3 and C receives 1/3. These shares total 1.

A's old share was (1/6) and he sacrifices (1/6) so A becomes zero. B has (4/6) and C has (2/6).