New Ratios

Taking share in old ratio does not change the mutual ratio of continuing partners. So new ratio remains their old mutual ratio.

A’s old share was 4 out of 10, or 4/10. B and C acquire this share in the ratio 1:3, so B receives 1/10 and C receives 3/10. Therefore, B’s new share is 2/10 + 1/10 = 3/10, while C’s new share is 4/10 + 3/10 = 7/10. Hence, the new ratio is 3:7. Exam tip: divide the retiring partner’s share according to the gaining ratio and add each gain to the partner’s old share.

The total of the old ratio is 10 parts. A has 2/10, B has 6/10, and C has 2/10. When B retires, his 6/10 share is acquired equally by A and C, so each receives 3/10. A's new share becomes 2/10 + 3/10 = 5/10, and C's new share also becomes 2/10 + 3/10 = 5/10. Thus their new ratio is 5:5, which reduces to 1:1. Options B, C, and D are equivalent-looking unreduced ratios, but in an MCQ asking for the ratio in simplest form, option A is the proper answer. The calculation also confirms that all profits are still allocated: 5/10 + 5/10 = 1.

A’s share is 8/10, while B’s and C’s old shares are 1/10 each. B and C acquire A’s 8/10 share in their old mutual ratio of 1:1, so each receives an additional 4/10. Their new shares become 5/10 and 5/10, which simplify to 1:1. Exam tip: When the remaining partners acquire the outgoing partner’s share in their old mutual ratio, their new mutual ratio remains unchanged.

The greatest common divisor of the terms 4 and 6 is 2. Dividing both terms by 2 gives the simplified ratio 2:3, so option A is correct. Option B represents the same ratio but is not in its simplest form. In an exam, divide all terms of a ratio by their greatest common divisor to simplify it.

After A’s retirement, B and C share the profits in the new ratio 7:5. The total parts are 7+5=12, so B’s share is 7/12 of the total profit. Option B, 5/12, is C’s share, while 7:5 is the ratio itself, not B’s fractional share. Exam tip: Divide a partner’s ratio part by the sum of all parts in the new ratio.

The governing concept is comparison of old and new profit-sharing fractions. The old total is 5 + 4 + 1 = 10, so A originally had 5/10 and B had 4/10. After C retires, A's new share is 3/5, which is 6/10, so A gains 1/10. B's new share is 2/5, which is 4/10, exactly equal to his old share. Thus A alone receives the released share of C, while B neither gains nor loses. Option B is wrong because B does not gain, option C is wrong because the gains are unequal, and option D ignores A's increase. Therefore, option A is correct.

To identify the gaining partner, compare each continuing partner's old share with the new share. The old ratio 5:4:1 has a total of 10, giving A 5/10 and B 4/10. The new ratio 3:2 has a total of 5, giving A 3/5 = 6/10 and B 2/5 = 4/10. Therefore A's share rises by 1/10, while B's share remains unchanged. C's entire 1/10 share is effectively taken by A because B does not increase his fraction. Option B is incorrect because the partners do not gain equally; option C reverses the result; and option D is impossible because A's share increases. Hence only A gained, so option A is correct.

C’s retirement allows A and B to gain C’s share. A’s old share is 3/10 and his new share is 2/5, so A’s gain is 2/5 − 3/10 = 1/10. B’s old share is 4/10 and his new share is 3/5, so B’s gain is 3/5 − 4/10 = 2/10. Therefore, the gaining ratio is 1/10 : 2/10 = 1:2. Exam tip: Calculate each partner’s gain by subtracting the old share from the new share, then simplify the ratio.

A gain is 5/7 - 6/10 = 4/35 and B gain is 2/7 - 1/10 = 13/70. With common denominator the ratio is 8:13 so it is not among options.

Under the old ratio, A's share was 6/10 and B's share was 1/10. Under the new ratio, A's share is 5/7 and B's share is 2/7. Therefore, A's gain is 5/7 − 6/10 = 8/70, while B's gain is 2/7 − 1/10 = 13/70. Hence, the gaining ratio is 8:13. The ratio 5:2 is the new profit-sharing ratio, not the gaining ratio. Exam tip: Calculate each partner's gain by subtracting the old share from the new share.

Before B’s retirement, the old mutual ratio of A and C was 1:3. Since A and C acquire B’s share in the same 1:3 ratio, their new relative profit-sharing ratio remains 1:3. Verification: A’s new share is 1/10 + (1/3 × 6/10) = 1/4, while C’s new share is 3/10 + (3/4 × 6/10) = 3/4; hence the ratio is 1:3. Exam tip: If the continuing partners acquire the retiring partner’s share in their old mutual ratio, their mutual ratio remains unchanged.

Direct answer: Option A, 33:17. The total ratio is 5+2+3=10. A has 5/10, B has 2/10 and C has 3/10. B’s share of 2/10 is taken by A and C in 4:1, a total of five parts. A receives (2/10)×(4/5)=4/25, while C receives (2/10)×(1/5)=1/25. Convert to denominator 50: A’s old share is 25/50 and the gain is 8/50, giving 33/50. C’s old share is 15/50 and the gain is 2/50, giving 17/50. Thus the new ratio is 33:17. Option A is correct. Option B, 5:3, is the ratio of A and C before B’s share is transferred. Option C, 9:4, is not obtained by adding the correct gains. Option D, 4:1, is only the gain-sharing ratio. Check: 33/50+17/50=1. Memory cue: the surviving partners keep their own shares and add the deceased partner’s share received by them.

The governing rule is that a retiring partner's share is added to the continuing partners in the stated gaining ratio. The old total is 3 + 7 + 5 = 15. A's retiring share is therefore 3/15. Since B and C acquire it equally, each receives half of 3/15, or 3/30. B's new share is 7/15 + 3/30 = 14/30 + 3/30 = 17/30. C's new share is 5/15 + 3/30 = 10/30 + 3/30 = 13/30. The common denominator cancels, giving the new ratio 17:13. The other options either retain the old ratio or divide the shares incorrectly. Hence option A is correct.

After retirement or death continuing partners share future profits and losses in new ratio. This is the main use in the chapter.

Gaining ratio shows how much extra share continuing partners got. Therefore it is used in goodwill adjustment.

If no special information is given continuing partners remain in their old mutual ratio. This is a common exam rule.

A's old share was \(\frac{2}{10}\). After C's retirement, the new ratio is 1:1, so A's new share is \(\frac{1}{2}=\frac{5}{10}\). Therefore, A's gain = new share − old share = \(\frac{5}{10}-\frac{2}{10}=\frac{3}{10}\). Option C represents only A's old share, not the gain. Exam tip: Calculate a partner's gain by subtracting the old share from the new share.

B’s old profit share was 1/(2+1+7) = 1/10. Under the new 1:1 ratio, B’s share becomes 1/2 = 5/10. Therefore, B’s gain = new share − old share = 5/10 − 1/10 = 4/10, or 2/5. The option 1/2 represents B’s new share, not the gain. Exam tip: calculate gain as the increase in share, namely new share minus old share.

After C retires, the combined old share of A and B is 2/10 + 1/10 = 3/10. Under the new 1:1 ratio, each receives 3/20. A’s old share was 2/10 = 4/20, so A sacrifices 1/20. B’s old share was 1/10 = 2/20, so B gains 1/20. Therefore, only B gains, while A sacrifices; a 3:4 gaining ratio between A and B is not formed. Exam tip: subtract the old share from the new share— a positive result indicates gain and a negative result indicates sacrifice.

When no information is given about the acquisition of the retiring partner’s share, the continuing partners acquire that share in their old profit-sharing ratio. Since A, B and C had an old mutual ratio of 3:2:3, their new ratio remains 3:2:3. Option B incorrectly changes C’s share without any basis. Exam tip: In the absence of specific information, retain the old mutual ratio of the continuing partners.

Direct answer: Option A, 4:3:2. The old ratio of A, B, C and D is 4:1:3:2. B retires, so B’s term is removed when we ask for the continuing partners’ old mutual ratio. The remaining partners are A, C and D, and their old shares are still 4, 3 and 2 units. Therefore their mutual ratio is 4:3:2. Option A is correct. Option B, 4:1:3, wrongly keeps B’s share and omits D. Option C, 1:3:2, begins with B’s old share, so it is not the ratio of A, C and D. Option D, 3:2:4, changes the order and values without any stated reason. No gaining or sacrificing information is supplied, so there is no further redistribution to calculate. The word “mutual” means compare only the partners who continue, using their old relative shares. Exam cue: delete the retiring partner’s number and preserve the order of the remaining partners.

The total ratio is 10, so D’s old share is 4/10. Since D’s share is acquired equally by A, B and C, each receives 4/30. A’s old share was 1/10 = 3/30. Therefore, A’s new share is 3/30 + 4/30 = 7/30. Exam tip: first convert the old ratio into fractional shares, then add the acquired share.

Under the old ratio, A’s share was 8/10 and B’s share was 1/10. Under the new ratio, A’s share becomes 9/10 while B’s share remains 1/10. Therefore, A gained 1/10 and B’s share remained unchanged; hence, only A gained a share. Exam tip: Gain = New Share − Old Share.

C share is 1/10 and when divided in 1:2:1 A gets 1/40 B gets 2/40 and D gets 1/40. New shares are 9/40 14/40 and 17/40 so the ratio is 9:14:17.