New Ratios

B’s old share was \(\frac{3}{4+3+2}=\frac{3}{9}\). After C’s death, the new ratio of A and B is 5:4, so B’s new share is \(\frac{4}{5+4}=\frac{4}{9}\). Therefore, B’s gaining share = new share − old share = \(\frac{4}{9}-\frac{3}{9}=\frac{1}{9}\). Option C is only B’s old share, not the gain. Exam tip: Calculate a gaining share as the new share minus the old share.

Under the old ratio 3:5:2, A’s share was 3/10 and B’s share was 5/10. Under the new ratio 2:3, A’s share becomes 2/5 = 4/10 and B’s share becomes 3/5 = 6/10. Thus, A gains 4/10 − 3/10 = 1/10, while B gains 6/10 − 5/10 = 1/10. Therefore, their gaining ratio is 1/10:1/10 = 1:1. Remember that 2:3 is the new ratio, not the gaining ratio.

The old shares are A = 2/10, B = 3/10 and C = 5/10. When C retires, C’s 5/10 share is acquired by A and B in the ratio 3:2. Hence A receives 3/10 and B receives 2/10 from C’s share. Their new shares become A = 2/10 + 3/10 = 5/10 and B = 3/10 + 2/10 = 5/10.

Both continuing partners therefore receive equal portions of future profits. The final ratio is 5:5, which reduces to 1:1. The ratio 3:2 is only the acquisition ratio and cannot be used alone as the new ratio. Thus option A, 1:1, is correct; option D expresses the same ratio before simplification but is not the selected answer.

C’s original share is \(3/10\). Since A and B acquire it in the ratio 1:2, A receives \(1/10\) and B receives \(2/10\). A’s new share is \(3/10+1/10=4/10\), while B’s new share is \(4/10+2/10=6/10\). Therefore, the new ratio is \(4:6=2:3\). Remember that 1:2 is the ratio in which the retiring partner’s share is acquired, not the new profit-sharing ratio.

A’s original share is \(2/10=1/5\). B acquires \(2/3\) of A’s share, i.e. \(2/15\), while C acquires \(1/3\), i.e. \(1/15\). Thus, B’s new share is \(5/10+2/15=19/30\), and C’s new share is \(3/10+1/15=11/30\). Therefore, the new ratio is \(19/30:11/30=19:11\). Exam tip: First calculate the retiring partner’s share, then distribute it among the gaining partners in the stated gaining ratio.

The direct answer is option A, 17:3. Take the total profit as 10 equal parts. A has 8/10, B has 1/10 and C has 1/10. B dies, and A and C take B's 1/10 equally. Half of 1/10 is 1/20, so A receives 1/20 and C also receives 1/20. A's new share is 8/10 + 1/20 = 16/20 + 1/20 = 17/20. C's new share is 1/10 + 1/20 = 2/20 + 1/20 = 3/20. Removing the common denominator 20 gives the new ratio 17:3. Option A is correct. Option B, 8:1, merely keeps A's old part and C's old part and ignores the transferred share. Option C, 9:2, incorrectly adds whole parts instead of half of B's share. Option D, 1:1, ignores that A already had a much larger share. Exam cue: first convert the old ratio into fractions, divide the deceased partner's share as instructed, and add each gain to the continuing partner's old share.

The direct answer is option A, 2:3:1. The old ratio is A:B:C:D = 2:3:1:4. When D retires, only D leaves; the question gives no special instruction about how D's share will be acquired. In this standard partnership-accounting situation, the continuing partners' old mutual ratio remains unchanged unless another ratio is specifically stated. Therefore A, B and C continue in the ratio 2:3:1. Option A is correct. Option B, 2:3:4, incorrectly includes D's four parts even though D has retired. Option C, 3:1:4, changes the order and again includes D, so it cannot be the ratio of A, B and C after D's retirement. Option D, 1:1:1, assumes equal sharing, which is not given and conflicts with the old mutual ratio. The total old parts were 10, but the common factor is not important; after D's share is dealt with without any stated change, the relative shares of A, B and C remain 2, 3 and 1. Memory cue: no information means preserve the continuing partners' old mutual ratio.

D’s old share was 4 out of 10, i.e. 4/10. Since this share is acquired equally by A, B and C, each receives 4/30. Therefore, their new shares are 1/10 + 4/30 = 7/30, 2/10 + 4/30 = 10/30, and 3/10 + 4/30 = 13/30. Hence, the new ratio is 7:10:13. Exam tip: When the retiring partner’s share is acquired equally, add an equal fraction of that share to each continuing partner’s old share.

B’s share is 2/10. Dividing it in the ratio 1:2:1 gives A an additional 1/20, C an additional 1/10, and D an additional 1/20. Therefore, the new shares are A = 3/10 + 1/20 = 7/20, C = 4/10 + 1/10 = 10/20, and D = 1/10 + 1/20 = 3/20. Hence, the new ratio is 7:10:3. The ratio 1:2:1 represents only the distribution of B’s acquired share, not the final profit-sharing ratio. Exam tip: First calculate the retiring partner’s share, then distribute it according to the stated acquisition ratio.

C share is 3/10. A gets 1/10 B gets 3/20 and D gets 1/20 so new shares are 10/20 5/20 and 5/20.

When share is taken in old ratio the mutual ratio of continuing partners remains unchanged. Therefore A B C new ratio is 5:2:2.

Direct answer: Option A, 9:3:8. The total old ratio is 2+4+1+3=10. A’s share is 2/10. This share is taken by B, C and D in 1:1:2, whose total is 4. B receives (2/10)×(1/4)=1/20; C also receives 1/20; D receives (2/10)×(2/4)=1/10. B’s new share is 4/10+1/20=8/20+1/20=9/20. C’s new share is 1/10+1/20=2/20+1/20=3/20. D’s new share is 3/10+1/10=6/20+2/20=8/20. Thus the ratio is 9:3:8. Option A is correct. Option B, 4:1:3, is only the old ratio of B, C and D. Option C, 1:1:2, is their receiving ratio, not the final ratio. Option D, 8:2:6, is an incorrect scaling and totals only 16 units rather than 20. Check that 9+3+8=20.

After retirement only continuing partners run the firm. Therefore future profits and losses are shared in new ratio.

Gaining ratio shows who received extra profit share. Therefore goodwill compensation is made in this ratio.

Without extra information continuing partners remain in their old mutual ratio. This is the common exam rule.

Gaining ratio is the same ratio in which continuing partners acquire extra share. Hence it will be 4:7.

When new ratio is clearly given future profits and losses are shared in that ratio. Gaining ratio only shows extra share.

When a continuing partner’s new share is lower than his old share, he gives up a part of his profit share. Therefore, sacrifice = old share − new share. Gain applies when the new share is greater than the old share. In examinations, an increase in share indicates gain, while a decrease indicates sacrifice.

Under the old ratio, A’s share was 2/10 and B’s share was 3/10. Under the new 5:5 ratio, each receives 5/10. Therefore, A’s gain is 5/10 − 2/10 = 3/10, while B’s gain is 5/10 − 3/10 = 2/10. Hence, the gaining ratio is 3:2. Exam tip: Calculate each partner’s gain as new share minus old share; the new ratio itself is not the gaining ratio.

Future profits are shared in new ratio. Gaining ratio is used only for extra share and goodwill adjustment.

C share is 4/15. Dividing it in 5:3 makes A and B new shares 7/10 and 3/10 so the ratio is 7:3.

B’s share is 2/10. A and C acquire it in the ratio 1:4, so A gains 2/10 × 1/5 = 1/25 and C gains 2/10 × 4/5 = 4/25. A’s new share is 4/10 + 1/25 = 11/25, while C’s new share is 4/10 + 4/25 = 14/25. Therefore, the new ratio is 11:14. Exam tip: First divide the deceased partner’s share according to the gaining ratio, then add each gain to the partner’s old share.

The total old ratio is 9+5+1=15, so A’s old share is 9/15. After C’s retirement, A’s new share is 2/(2+1)=2/3=10/15. Therefore, A’s gaining share = new share − old share = 10/15−9/15=1/15. Option C represents A’s old share, not the share gained. Exam tip: Always calculate the gain by subtracting the old share from the new share.

B old share is 5/15 and new share is 1/3 or 5/15. There is no change so no gain or sacrifice occurs.

A share increased from 9/15 to 10/15 and B share remained 5/15. Therefore only A gained C share.