New Ratios

Remaining partners bear retiring partner goodwill based on increased share. Therefore gaining ratio is useful in goodwill adjustment.

The relevant principle is that the retiring partner’s share is distributed according to the stated gaining ratio and then added to each continuing partner’s old share. The total old ratio is 10, so D’s share is 1/10. Since A, B and C acquire it in 3:2:1, the total parts are 6 and C receives 1/6 of D’s share, namely 1/60. C’s old share was 2/10, or 12/60. Therefore C’s new share is 12/60 + 1/60 = 13/60. Thus option A is correct. Option C represents only the additional share obtained from D, not C’s complete new share; options B and D do not apply the stated acquisition ratio correctly.

Gaining ratio is determined by comparing each continuing partner’s new share with his or her old share. The old total is 10, so A had 2/10 and B had 3/10. After C’s retirement, the new ratio 5:5 means each receives 5/10. A’s gain is 5/10 − 2/10 = 3/10, while B’s gain is 5/10 − 3/10 = 2/10. The common factor 1/10 can be omitted, giving a gaining ratio of 3:2. Therefore option A is correct. Options B and C merely describe the new ratio, not the increase in shares. Option D reverses the actual gains of A and B.

To determine who gained, compare the old and new fractions rather than comparing the ratio numbers directly. The old total is 10, so A held 6/10 and B held 3/10. After C’s retirement, the new ratio 5:4 totals 9; therefore A receives 5/9 and B receives 4/9. A’s change is 5/9 − 6/10 = 50/90 − 54/90 = −4/90, which is a sacrifice, not a gain. B’s change is 4/9 − 3/10 = 40/90 − 27/90 = 13/90, which is a gain. Thus B alone gains, and B has gained more share. Option A is correct. Options B, C and D contradict the calculated changes.

The governing idea is that sacrifice or gain equals new share minus old share. The old total is 10, so A’s old share is 6/10 = 3/5. The continuing partners’ new ratio is 5:4, giving A a new share of 5/9. Using denominator 45, the old share is 27/45 and the new share is 25/45. Thus the change is 25/45 − 27/45 = −2/45. The negative sign means A has sacrificed 2/45 of the total profit. Therefore option A is correct. Option B reverses the direction, option C ignores the difference, and option D uses an unsupported amount.

The concept is the computation of a surviving partner’s new share when a deceased partner’s share is acquired in a specified ratio. The old total is 10, so A’s share is 1/10. B, C and D acquire it in 1:2:2, whose total is 5; D therefore receives 2/5 × 1/10 = 1/25. D’s old share was 4/10 = 10/25. Adding the acquired portion gives D’s new share as 10/25 + 1/25 = 11/25. Hence option A is correct. Option B is only D’s old share, option C is not D’s acquired portion, and option D equals 1/2, which is greater than the correctly calculated share.

To analyse gain or sacrifice after a partner’s death, compare each continuing partner’s old share with the new share. The old ratio 2:3:5 has 10 parts, so B’s old share was 3/10 and C’s old share was 5/10. The new ratio 5:5 gives each continuing partner 5/10. B therefore gains 5/10 − 3/10 = 2/10. C’s share remains 5/10, so C neither gains nor sacrifices. Hence option A is the only correct statement. The distractor saying C sacrificed ignores that C’s new share equals the old share; the remaining options also incorrectly describe the unchanged share or deny B’s gain.

C share of 2/15 is divided in 2:3. New shares of A and B are 44/75 and 31/75 so the ratio is 44:31.

The total of the original ratio is 18, so B’s share is 7/18. A receives 4/7 of B’s share (4/18), and C receives 3/7 (3/18). Their new shares are therefore A: 6/18 + 4/18 = 10/18 and C: 5/18 + 3/18 = 8/18. The new ratio is 10:8, or 5:4.

The original shares are out of 15 parts, so A’s share is 9/15. B receives 5/9 of A’s share, or 5/15; C receives 4/9, or 4/15. Adding these gains to their original shares gives B: 4/15 + 5/15 = 9/15 and C: 2/15 + 4/15 = 6/15. Thus, their new ratio is 9:6, which simplifies to 3:2.

A’s old profit share is \(\frac{7}{7+6+5}=\frac{7}{18}\). After C’s retirement, A’s new share is \(\frac{5}{5+4}=\frac{5}{9}=\frac{10}{18}\). Therefore, A’s gaining share is \(\frac{10}{18}-\frac{7}{18}=\frac{3}{18}=\frac{1}{6}\). Exam tip: Gaining share is calculated as new share minus old share.

B’s old profit share is 6/(7+6+5) = 6/18 = 1/3. Under the new ratio 5:4, B’s share is 4/9. Therefore, B’s gaining share = new share − old share = 4/9 − 1/3 = 4/9 − 3/9 = 1/9. In such questions, first calculate each partner’s old and new fractional share, then subtract the old share from the new share.

The total of the old ratio 7:6:5 is 18, so A and B had old shares of 7/18 and 6/18. Under the new ratio 5:4, their shares become 5/9 = 10/18 and 4/9 = 8/18. Therefore, A’s gain is 10/18 − 7/18 = 3/18 = 1/6, while B’s gain is 8/18 − 6/18 = 2/18 = 1/9. Hence, the gaining ratio is 1/6 : 1/9 = 3:2. Exam tip: calculate each partner’s gain as new share minus old share before forming the ratio.

The old total ratio is 6+5+4=15. Therefore, A's old share was \(6/15=2/5\), while A's new share after C's retirement is \(1/(1+2)=1/3=5/15\). Since \(5/15<6/15\), A's share has decreased. In contrast, B's share has increased from \(5/15\) to \(2/3=10/15\). Exam tip: Convert the old and new shares to a common denominator before comparing them.

B’s old profit share was \(5/(6+5+4)=5/15=1/3\). In the new ratio 1:2, B’s new share is \(2/3\). Therefore, B’s gaining share = new share − old share = \(2/3-1/3=1/3\). Option B is only B’s new share, not the gain. Exam tip: For a retiring partner’s reconstitution, calculate a continuing partner’s gain by subtracting the old share from the new share.

Under the old ratio, A’s share was \(6/15=2/5\) and B’s share was \(5/15=1/3\). Under the new ratio, A receives \(1/3\) and B receives \(2/3\). Thus, A’s share decreases by \(1/15\), while B’s share increases by \(1/3\). Therefore, only B gains, so the gaining ratio of A:B is 0:1. Exam tip: Calculate each partner’s gain as new share minus old share; a negative result indicates a sacrifice, not a gain.

The new ratio is found by adding each continuing partner’s acquired portion to his or her old share. The old total is 10, so B’s retiring share is 2/10 = 1/5. A acquires 3/5 of this, equal to 3/25, while C acquires 2/5, equal to 2/25. A’s new share is 5/10 + 3/25 = 25/50 + 6/50 = 31/50. C’s new share is 3/10 + 2/25 = 15/50 + 4/50 = 19/50. Since the common denominator cancels, the new ratio is 31:19. Therefore option A is correct. The other choices either retain the old ratio or confuse the acquisition ratio with the final profit-sharing ratio.

A’s share is 7 out of the total 14 parts, i.e. 7/14. B receives 2/7 of A’s share, equal to 2/14, while C receives 5/7 of A’s share, equal to 5/14. Therefore, B’s new share is 4/14 + 2/14 = 6/14, and C’s new share is 3/14 + 5/14 = 8/14. Thus, the new ratio is 6:8 = 3:4. Exam tip: add each partner’s acquired share to their original share and reduce the resulting ratio.

The direct answer is option A, 1:2. A, B and C initially share profit in 3:4:5, so their total parts are 12 and A's share is 3/12. When A retires, the question directly says that B and C acquire A's share in the ratio 1:2. Therefore, B gains one part and C gains two parts; the gaining ratio is 1:2. Option A is correct because it states exactly this additional-share ratio. Option B, 4:5, is the old profit-sharing ratio of B and C, not the ratio in which they gain A's share. Option C, 3:4, uses A's old part and B's old part and does not describe the extra benefit received by B and C. Option D, 1:1, would mean that both partners took equal portions, which the question does not say. The old ratio is needed for finding final shares, but here the gaining ratio is already supplied. Memory cue: gaining ratio means the ratio in which continuing partners take the retiring partner's share.

When remaining partners continue in old ratio the extra share is also acquired in that ratio. Here A B ratio is 10:7.

After B retires, only A and C will share the profits. Their old relative shares were 13 for A and 7 for C, so their new ratio remains 13:7. The ratio 13:5 is incorrect because it includes B’s old share. Exam tip: when the continuing partners retain their old relative ratio, use the corresponding terms of their original ratio directly as the new ratio.

After A’s death, only B and C will share the profits. Their existing mutual ratio is 8:10, which simplifies to 4:5 after dividing both terms by 2. Therefore, option A is correct. In an examination, always express the ratio in its simplest form; 8:10 is equivalent but not the simplest form.

A’s old share was \(5/(5+9+6)=5/20=1/4\). Under the new ratio 4:5, A’s new share is \(4/(4+5)=4/9\). Therefore, A’s gaining share = new share − old share = \(4/9-1/4=(16-9)/36=7/36\). Option B is only the new share, not the gaining share. In an exam, use ‘new share − old share’ to calculate a partner’s gain.

B’s old profit share was \(9/(5+9+6)=9/20\). Under the new ratio 4:5, B’s new share is \(5/(4+5)=5/9\). Therefore, B’s gaining share = new share − old share = \(5/9-9/20=(100-81)/180=19/180\). Exam tip: calculate the gaining share by subtracting the old share from the new share.

Under the old ratio, A’s share was \(5/20=1/4\) and B’s share was \(9/20\). Under the new ratio, A’s share is \(4/9\) and B’s share is \(5/9\). Therefore, A’s gain is \(4/9-1/4=7/36=35/180\), while B’s gain is \(5/9-9/20=19/180\). Hence, the gaining ratio is \(35:19\). Option 4:5 is the new profit-sharing ratio, not the gaining ratio. Exam tip: Gaining ratio = new share − old share.