New Profit Sharing Ratio

After C receives 1/5, the combined share left for A and B is 4/5. Simplify the old ratio 9:6 to 3:2. Therefore, A’s new share is 4/5 × 3/5 = 12/25, and B’s new share is 4/5 × 2/5 = 8/25. C’s share is 1/5 = 5/25. Hence, the new ratio is 12:8:5. Option B incorrectly adds C’s share to the old ratio without reducing A’s and B’s shares. Exam tip: When the incoming partner’s share is sacrificed by the old partners in their old ratio, distribute the remaining share, 1 minus the incoming partner’s share, in that old ratio.

The direct answer is A: A’s sacrifice is \(1/15\). The old ratio 7:8 means A originally had \(7/15\), while B had \(8/15\). C receives \(1/3\), so the two old partners together retain \(1-1/3=2/3\). Their new mutual ratio is 3:2, whose total is 5; therefore A’s new share is \(2/3\times3/5=2/5\). Sacrifice means old share minus new share: \(7/15-2/5=7/15-6/15=1/15\). Option A is correct. Option B, \(7/15\), is A’s old share, not sacrifice. Option C, \(2/15\), is not the difference. Option D says gain, but A gives up part of his share, so it is sacrifice, not gain. Remember: sacrifice = old share − new share.

After C’s admission, the profit share left for A and B is \(1-\frac{1}{3}=\frac{2}{3}\). Under the new ratio 3:2, B’s new share is \(\frac{2}{3}\times\frac{2}{5}=\frac{4}{15}\). B’s old share was \(\frac{8}{15}\), so B’s sacrifice is \(\text{old share}-\text{new share}=\frac{8}{15}-\frac{4}{15}=\frac{4}{15}\). The exam rule is to subtract the incoming partner’s reduced share from the old partner’s share, not the reverse.

Old partners retain (5/7) and B's new share is (5/7\times4/5=4/7). B's old share was (2/7) so B gains.

Since C receives the entire share from A, only A sacrifices profit and B’s share stays intact. Exam tip: first write \(A'=A-C\) and \(B'=B\); proportional reduction applies only when all old partners sacrifice.

C’s incoming share is \(\frac{1}{4}\), so the total sacrifice made by A and B must also be \(\frac{1}{4}\). Therefore, B’s sacrifice = total sacrifice − A’s sacrifice = \(\frac{1}{4}-\frac{1}{8}=\frac{1}{8}\). Option C is incorrect because it would make the total sacrifice only \(\frac{3}{16}\). Exam tip: The incoming partner’s share equals the total sacrifice of the old partners.

C receives 1/4 of the profits, so A and B each sacrifice 1/8 because the sacrifice is equal. A’s new share is 3/4 − 1/8 = 5/8, B’s new share is 1/4 − 1/8 = 1/8, and C’s share is 1/4 = 2/8. Therefore, the new ratio is 5/8 : 1/8 : 2/8 = 5 : 1 : 2. Exam tip: When the sacrifice is equal, subtract an equal portion from each old partner’s share.

D's (1/5) share comes from A and B in (3:1) so A sacrifices (3/20) and B (1/20). New shares are (9/20) (5/20) (2/20) and (4/20).

The direct answer is A: B’s sacrifice is \(2/9\). From the old ratio 4:5, A’s old share is \(4/9\) and B’s is \(5/9\). C receives \(1/3\), so A and B together retain \(2/3\). Their new mutual ratio is 1:1, so each receives half of \(2/3\), namely \(1/3\). B’s sacrifice is therefore \(5/9-1/3=5/9-3/9=2/9\). Option A matches this result. Option B, \(1/9\), is too small and is not the difference. Option C, \(1/3\), is B’s new share, not his sacrifice. Option D, \(5/9\), is B’s old share. Sacrifice measures the part surrendered, not the final share. Exam cue: first calculate the old share, then the new share, and subtract new from old.

A’s old share was \(4/9\). After C receives \(1/3\), the remaining \(2/3\) profit is shared equally by A and B, so A’s new share is \(1/3\). Therefore, A’s sacrifice = old share − new share = \(4/9-1/3=1/9\). Option C represents A’s new share, not the sacrifice. Exam tip: calculate sacrifice by subtracting the new share from the old share.

C receives 1/3 of the profits, so the remaining share for A and B is 2/3. Dividing this remainder in the ratio 2:1 gives A's share as (2/3) × (2/3) = 4/9 and B's share as (2/3) × (1/3) = 2/9. C's share is 1/3 = 3/9. Therefore, the new ratio is 4:2:3. The original 5:1 ratio is not used directly because the question specifies that the remaining profit will be shared in the ratio 2:1. Exam tip: subtract the admitted partner's share from 1 first, then distribute the balance according to the stated ratio of the old partners.

C receives 1/3, leaving 2/3 for A and B. Therefore, A’s new share is \(\frac{2}{3}\times\frac{2}{3}=\frac{4}{9}\), while B’s new share is \(\frac{2}{3}\times\frac{1}{3}=\frac{2}{9}\). B’s old share was \(\frac{1}{6}\), so B gains \(\frac{2}{9}-\frac{1}{6}=\frac{1}{18}\). A’s new share, \(\frac{4}{9}\), is lower than A’s old share of \(\frac{5}{6}\), so A sacrifices rather than gains. Exam tip: compare each old partner’s new share with the old share; a positive difference indicates a gain.

A becomes (7/9-1/9=6/9) and B remains (2/9). The old partners' new mutual ratio should be (6:2=3:1).

In the original ratio 7:2, the total is 9 parts, so A’s share is 7/9 and B’s share is 2/9. Since C’s 1/9 share is taken entirely from A, A’s new share becomes 7/9 − 1/9 = 6/9, while B’s share remains 2/9. Therefore, A and B’s new mutual ratio is 6:2 = 3:1. The ratio 7:2 is the old ratio, not the new one. Exam tip: Reduce the share only of the partner who sacrifices the share for the incoming partner.

C’s required share is \(\frac{1}{4}\). Total sacrifice by A and B is also \(\frac{1}{4}\). Since sacrifice ratio \(A:B=3:2\), B sacrifices \(\frac{2}{3+2}\times \frac{1}{4}=\frac{2}{5}\times \frac{1}{4}=\frac{2}{20}=\frac{1}{10}\). B’s original share is \(\frac{7}{13+7}=\frac{7}{20}\). So B’s new share \(=\frac{7}{20}-\frac{1}{10}=\frac{7}{20}-\frac{2}{20}=\frac{5}{20}=\frac{1}{4}\).

C's share is \(1/4=5/20\). Since A and B sacrifice in the ratio 3:2, A sacrifices \(3/20\) and B sacrifices \(2/20\). Thus, A's new share is \(13/20-3/20=10/20\), B's is \(7/20-2/20=5/20\), and C's share is \(5/20\). Therefore, the new ratio is \(10:5:5\), or \(2:1:1\) in simplest form. Exam tip: Distribute the incoming partner's share among the old partners according to the sacrificing ratio and subtract those amounts from their old shares.

The old shares are A = 4/10, B = 4/10 and C = 2/10. D’s 1/10 share is taken equally from A and C, so each gives up 1/20. The new shares are A = 4/10 − 1/20 = 7/20, B = 4/10 = 8/20, C = 2/10 − 1/20 = 3/20 and D = 1/10 = 2/20. Thus, the new ratio is 7:8:3:2.

The sacrificing ratio is based on the shares surrendered by the old partners. A gives up 1/16 and B also gives up 1/16, so their sacrificing ratio is 1/16:1/16 = 1:1. Together, their sacrifices equal C’s 1/8 share.

Express the old ratio 5:3 with denominator 16: A's share is 10/16 and B's share is 6/16. C receives 1/8 = 2/16. Since A and B sacrifice this share equally, each sacrifices 1/16. Therefore, A's new share is 10/16 − 1/16 = 9/16, B's is 6/16 − 1/16 = 5/16, and C's is 2/16. Hence, the new ratio is 9:5:2. Exam tip: The total sacrifice by the old partners equals the new partner's share; when sacrifice is equal, divide it equally among them.

B’s new share is 1 − 1/3 − 1/5 = 7/15. B’s old share was 3/5 = 9/15, so B’s sacrifice is old share minus new share: 9/15 − 7/15 = 2/15.

A’s old share is 2/5 and his new share is 1/3, so A’s sacrifice = 2/5 − 1/3 = 1/15. Since C receives 1/5, B’s new share is 1 − 1/3 − 1/5 = 7/15. Therefore, B’s sacrifice = 3/5 − 7/15 = 2/15. Hence, the sacrificing ratio is 1/15 : 2/15 = 1:2. In the examination, calculate sacrifice as old share minus new share.

The total of the new profit shares is 1. Therefore, A’s new share = 1 − B’s new share − C’s share = 1 − 1/2 − 1/8 = 3/8. A’s old share was also 3/8, so A neither sacrifices nor gains any share. Exam tip: Always verify that the new profit shares add up to 1.

A’s old share was 3/(3 + 5) = 3/8. A’s new share is the remainder after C’s and B’s shares: 1 − 1/8 − 1/2 = 3/8. Thus A’s sacrifice, calculated as old share minus new share, is 3/8 − 3/8 = 0.

Before C’s admission, A’s share was 3/8 and B’s share was 5/8. After admission, B’s share becomes 1/2, or 4/8, so B sacrifices 1/8. Since the total of the new shares is 1, A’s share remains 3/8; therefore, C’s 1/8 share comes entirely from B. Exam tip: Compare each existing partner’s old and new share to identify the sacrificing partner.

D’s 1/6 share is taken from B and C in the ratio 1:2. Therefore, B’s sacrifice is 1/6 × 1/3 = 1/18, and C’s sacrifice is 1/6 × 2/3 = 1/9. Expressing the old ratio in eighteenths gives A = 3/18, B = 6/18 and C = 9/18. The new shares are A = 3/18, B = 5/18, C = 7/18 and D = 3/18. Hence, the new ratio is 3:5:7:3. Exam tip: Deduct the incoming partner’s share from the old partners according to their sacrifice ratio.