New Ratios

The correct answer is A: equal. The gaining ratio shows how the continuing partners acquire the retiring partner’s share. If the question says they take that share equally, each partner receives the same additional fraction. Therefore their gains are equal, so the gaining ratio is equal. Option A directly follows from the stated condition. Option B, the old ratio, is not automatically the gain ratio; it may be different from the way the retiring share is acquired. Option C, the new ratio, describes final profit shares, not necessarily the amounts gained. Option D, the capital ratio, concerns capital balances and is irrelevant unless specifically stated. Exam cue: “taken equally” means “equal gaining ratio.”

When the continuing partners acquire the retiring partner’s share in their old mutual ratio, each gains in proportion to their existing shares. Their mutual ratio therefore stays unchanged, so the new ratio is the old mutual ratio of the continuing partners.

A’s share is 5/10. B acquires 2/5 of it, or 2/10, and C acquires 3/5 of it, or 3/10. Their new shares are therefore B: 2/10 + 2/10 = 4/10 and C: 3/10 + 3/10 = 6/10. Thus, their new ratio is 4:6, which simplifies to 2:3.

The correct answer is Option A, 11:9. The old ratio A:B:C is 3:5:2, so their fractions are 3/10, 5/10 and 2/10. B retires and gives up 5/10. A and C acquire this surrendered share equally, so each receives 5/20, which is 1/4 or 2.5/10. A's new share is 3/10 + 2.5/10 = 5.5/10 = 11/20. C's new share is 2/10 + 2.5/10 = 4.5/10 = 9/20. Therefore the new ratio is 11:9. Option A is correct. Option B, 3:2, does not reflect the equal acquisition of B's share. Option C, 5:4, is close in appearance but does not result from the stated fractions. Option D, 1:1, would mean A and C receive equal total profits, whereas A already had a larger original share. A useful method is to convert all shares to a common denominator before adding the gain. Exam cue: new share equals old share plus acquired share; divide only the retiring partner's surrendered share according to the stated gaining ratio.

The governing concept is the new profit-sharing ratio after the death of a partner. The old shares of B and C are 2/10 and 2/10, so their old mutual ratio is 2:2, or 1:1. A's share is 6/10. Since B and C acquire this entire share in the same 1:1 proportion, each receives 3/10 from A. Therefore, B's new share is 2/10 + 3/10 = 5/10, and C's new share is also 2/10 + 3/10 = 5/10. Thus the new ratio is 5:5, reduced to 1:1. Option B represents the unreduced equivalent, while C and D do not follow the stated acquisition rule.

After C retires, only A and B share the profit. In the new ratio 3:2, the total parts are 3+2=5, so A's new share is \(\frac{3}{5}\). Option B, \(\frac{2}{5}\), represents B's share. Exam tip: To find a partner's share from a ratio, divide that partner's ratio part by the sum of all ratio parts.

After C’s retirement, the new ratio of A and B is given as 4:6. Dividing both terms by their highest common factor, 2, gives 4:6 = 2:3. Therefore, 2:3 is correct. Option 4:6 is the unsimplified form, not the simplest form. In exams, reduce both terms by their highest common factor.

After A’s retirement, B and C share profits in the ratio 5:4. The total parts are 5+4=9, so B’s new share is 5/9. The fraction 4/9 represents C’s share, while 5/4 is the ratio itself, not an individual share. Exam tip: To find a partner’s share from a ratio, divide that partner’s ratio part by the sum of all ratio parts.

The direct answer is that A gained more share, so option A is correct. The original ratio 3:4:3 has total 10 parts, so A had 3/10 and B had 4/10. C retires, and the remaining total share is 1, which A and B now divide equally. Thus each receives 1/2, or 5/10. A’s gain is 5/10−3/10=2/10. B’s gain is 5/10−4/10=1/10. Since 2/10 is greater than 1/10, A gained more. Option A is correct. Option B is wrong because B gained, but by a smaller amount. Option C is wrong because their gains are not equal. Option D is wrong because both partners gained something when C’s share was redistributed and the new ratio became 1:1. Exam cue: compare new share minus old share, not merely the final shares.

A’s old share is 2/10 and its new share is 1/3, so A gains 1/3 − 1/5 = 2/15. B’s old share is 4/10 and its new share is 2/3, so B gains 2/3 − 2/5 = 4/15. Their gaining ratio is 2/15:4/15, or 1:2.

The total of the old ratio 2:4:4 is 10, so A and B’s old shares are 1/5 and 2/5 respectively. Under the new ratio 1:2, their shares become 1/3 and 2/3. Therefore, A’s gain is 1/3 − 1/5 = 2/15 and B’s gain is 2/3 − 2/5 = 4/15. Hence, the gaining ratio is 2/15:4/15 = 1:2. Exam tip: gaining ratio is calculated as new share minus old share.

Gaining ratio is calculated by subtracting each continuing partner's old share from the corresponding new share. A's old share is 4/10, while his new share under 1:1 is 1/2, or 5/10. Therefore, A's gain is 5/10 - 4/10 = 1/10. B's old share is 5/10, and his new share is also 1/2 = 5/10, so B's gain is zero. The gains are therefore 1/10 and 0, which give the gaining ratio 1:0. Option B reverses the partners' gains, option C incorrectly assumes both gained, and option D is merely the old ratio of A and B.

A and C have an old mutual ratio of 3:2. Since they acquire B’s share in this same 3:2 ratio, both partners’ shares increase proportionately, so their new mutual ratio remains 3:2. Verification: A’s new share is \(\frac{3}{6}+\frac{3}{5}\times\frac{1}{6}=\frac{3}{5}\), while C’s new share is \(\frac{2}{6}+\frac{2}{5}\times\frac{1}{6}=\frac{2}{5}\). Therefore, the new ratio is 3:2. Exam tip: If the continuing partners gain the retiring partner’s share in their old mutual ratio, their mutual ratio does not change.

The relevant principle is that the deceased partner’s entire old share is added to the continuing partners according to the stated acquisition ratio. The old total is 4+2+1=7, so A=4/7, B=2/7 and C=1/7. A receives 3/5 of B’s share, equal to 3/5×2/7=6/35. C receives 2/5×2/7=4/35. Thus A’s new share is 4/7+6/35=20/35+6/35=26/35, while C’s new share is 1/7+4/35=5/35+4/35=9/35. The common denominator gives 26:9, so option A is correct. Other options result from ignoring the old shares or applying 3:2 to the final shares.

A’s share is 2/10. B and C each acquire half of it, or 1/10. Their new shares are therefore B: 7/10 + 1/10 = 8/10 and C: 1/10 + 1/10 = 2/10. Thus, 8:2 simplifies to 4:1.

After retirement, the continuing partners carry on the firm. Their new ratio indicates how they will share future profits and, subject to their agreement, losses.

When no other information about the new ratio is given, the continuing partners are assumed to retain their old mutual ratio. Their shares are compared after excluding the retiring partner.

A’s old share was \(\frac{1}{1+4+5}=\frac{1}{10}\). After C’s retirement, A’s new share is \(\frac{3}{3+7}=\frac{3}{10}\). Therefore, A’s gain is \(\frac{3}{10}-\frac{1}{10}=\frac{2}{10}=\frac{1}{5}\). Exam tip: Gain is always calculated as new share minus old share.

B’s old share was \(\frac{4}{10}\), since the total of the old ratio is \(1+4+5=10\). In the new ratio 3:7, B’s new share is \(\frac{7}{10}\). Therefore, B’s gain = new share − old share = \(\frac{7}{10}-\frac{4}{10}=\frac{3}{10}\). Option D is only B’s old share, not the gain. Exam tip: always calculate gain as the new share minus the old share.

Under the old ratio, A’s share was 1/10 and B’s share was 4/10. Under the new ratio of 3:7, A’s share is 3/10 and B’s share is 7/10. Therefore, A’s gain is 3/10 − 1/10 = 1/5, while B’s gain is 7/10 − 4/10 = 3/10. Hence, the gaining ratio is 1/5 : 3/10 = 2:3. Exam tip: calculate each continuing partner’s gain as new share minus old share; do not confuse the new ratio 3:7 with the gaining ratio.

With no change to their mutual ratio specified, A, B, and C retain their old shares relative to one another: 2:2:3. D’s share is excluded.

Exclude B, who retires, and compare the old shares of the continuing partners in the order A, C, D: 3:2:1.

The total ratio is 10, so D’s share is 1/10. Since D’s share is acquired equally by three partners, each partner gets 1/30. A’s old share was 2/10; therefore, A’s new share is 2/10 + 1/30 = 6/30 + 1/30 = 7/30. Option B represents only A’s old share and ignores the additional share acquired from D. Exam tip: Add the retiring partner’s share to each continuing partner according to the agreed acquisition ratio.

D’s share is 1 out of the total 10 parts, that is, 1/10. Since this share is acquired equally by A, B and C, each receives 1/30. Their old shares are 2/10 = 6/30, 3/10 = 9/30 and 4/10 = 12/30. The new shares become 7/30, 10/30 and 13/30, giving the ratio 7:10:13. Exam tip: When a retiring partner’s share is acquired equally, add an equal increment to each continuing partner’s old share.

Under the old ratio, A’s share is 7/10 and B’s share is 2/10. Under the new ratio 4:1, A receives 4/5 = 8/10, so A gains 1/10. B receives 1/5 = 2/10, exactly the old share, so B is unchanged. Therefore, option A is correct; the other statements contradict these comparisons.