New Ratios

To determine a decrease, compare each continuing partner’s old share with the new share. The old total is 15, so A had 9/15 = 3/5 and B had 4/15. The new ratio 3:2 has total 5, so A’s new share is 3/5 and B’s new share is 2/5 = 6/15. Thus A’s share remains exactly 3/5; it has neither increased nor decreased. B’s share changes from 4/15 to 6/15, so B gains 2/15. C’s retirement share is absorbed by B. Therefore nobody’s share has decreased, making option D correct. Option A is tempting only if the old and new forms of A’s share are not simplified properly.

B’s old share was 4/(9+4+2) = 4/15. Under the new ratio of 3:2, B’s new share is 2/5 = 6/15. Therefore, B’s gaining share = new share − old share = 6/15 − 4/15 = 2/15. Option 2/5 is B’s new share, not the gain. In the exam, calculate a partner’s gaining ratio by subtracting the old share from the new share.

Under the old ratio 9:4:2, A, B and C had shares of 9/15, 4/15 and 2/15 respectively. In the new ratio 3:2, A’s new share is 3/5 = 9/15, so A gains nothing. B’s gain is 2/5 − 4/15 = 6/15 − 4/15 = 2/15. Therefore, the gaining ratio of A:B is 0:2/15 = 0:1. Exam tip: subtract each partner’s old share from the new share to identify the gain.

B’s old profit share is 3 out of 10, or 3/10. A acquires 2/3 of B’s share, which is 2/10, while C acquires 1/3, which is 1/10. Thus, A’s new share is 4/10 + 2/10 = 6/10 and C’s new share is 3/10 + 1/10 = 4/10. Therefore, the new ratio is 6:4 = 3:2. Remember that 2:1 is the gaining ratio, not the new profit-sharing ratio.

C’s share is \(2/15\). When this is acquired by A and B in the ratio 1:3, A receives \(1/4 \times 2/15=1/30\), while B receives \(3/4 \times 2/15=1/10\). Thus, A’s new share is \(11/15+1/30=23/30\), and B’s new share is \(2/15+1/10=7/30\). Therefore, the new ratio is \(23:7\). Exam tip: On retirement, distribute the outgoing partner’s share according to the gaining ratio, not the old profit-sharing ratio.

The governing procedure is to divide the retiring partner’s share according to the acquisition ratio and add those portions to the continuing partners’ old shares. The old total is 15, so A’s share is 2/15. B receives 3/4 of it: (3/4)(2/15) = 1/10. C receives 1/4 of it: (1/4)(2/15) = 1/30. B’s new share is 9/15 + 1/10 = 18/30 + 3/30 = 21/30. C’s new share is 4/15 + 1/30 = 8/30 + 1/30 = 9/30. Cancelling the common denominator gives 21:9. Therefore option A is correct; 3:1 is only the acquisition ratio, not the final ratio.

The old total ratio is 8+7+3=18, so B’s share is 7/18. A and C acquire this share in the ratio 5:2; therefore, A receives 5/18 and C receives 2/18. Their new shares are A = 8/18+5/18=13/18 and C = 3/18+2/18=5/18. Hence, the new ratio is 13:5. Exam tip: apply the acquisition ratio only to the outgoing partner’s share, not to the entire profit.

Gaining ratio is calculated as new share minus old share. The first partner’s gain is 1/2 − 7/20 = 10/20 − 7/20 = 3/20. The second partner’s gain is 1/2 − 1/4 = 2/4 − 1/4 = 1/4 = 5/20. Therefore, the gaining ratio is 3/20 : 5/20 = 3:5. The key exam tip is to compare each partner’s new share with their own old share; using only the new shares would incorrectly give 1:1.

Gaining share = new profit share − old profit share. Convert the old share to the common denominator: \(3/7=9/21\). Therefore, A’s gaining share is \(13/21-9/21=4/21\). Option B is only the new share and option C is only the old share, not the gain. Exam tip: Use a common denominator before subtracting the old share from the new share.

Gaining share = new share − old share. Therefore, B's gaining share is \(11/25-9/25=2/25\). Option C is the old share and option D is the new share, while option B results from an incorrect subtraction. Exam tip: Always subtract the old share from the new share to find the gaining share.

Gaining ratio is calculated from the increases in the partners’ shares, not from the final ratio itself. The new shares are 9/17 and 8/17. For the first partner, gain = 9/17 − 1/2 = (18 − 17)/34 = 1/34. For the second partner, gain = 8/17 − 1/3 = (24 − 17)/51 = 7/51. To compare these gains, use denominator 102: 1/34 = 3/102 and 7/51 = 14/102. Hence the gaining ratio is 3:14. Option A is correct. Options B and C confuse the new ratio or old-share relationship with the actual gains, while option D incorrectly assumes equal acquisition.

Under the old ratio 7:7:4, A and B each held 7/18. Under the new ratio 5:4, their new shares are 5/9 = 10/18 and 4/9 = 8/18. Therefore, A’s gain is 10/18 − 7/18 = 3/18, while B’s gain is 8/18 − 7/18 = 1/18. Hence, the gaining ratio is 3/18:1/18 = 3:1. Exam tip: calculate each continuing partner’s gain as new share minus old share, then simplify the ratio.

Before C’s retirement, A’s share was \(8/15\) and B’s share was \(2/15\). Under the new ratio 3:2, A’s share is \(3/5=9/15\) and B’s share is \(2/5=6/15\). Thus, both A and B have gained, so neither partner’s share has decreased. C’s share becomes nil after retirement, but the question asks only about A and B. Exam tip: Convert the old and new ratios to a common denominator before comparing the shares.

A's old share was \(8/15\). Under the new ratio 3:2, A's new share is \(3/5 = 9/15\). Therefore, A's gaining share = new share − old share = \(9/15 - 8/15 = 1/15\). Option B is only A's new share, not the gain. In the examination, calculate the gaining share by subtracting the old share from the new share.

B’s old share was \(2/(8+2+5)=2/15\). Under the new ratio of 3:2, B’s share is \(2/5=6/15\). Therefore, B’s gaining share is \(6/15-2/15=4/15\). The value 2/5 is B’s new share, not the gain. Exam tip: Gaining share = New share − Old share.

Under the old ratio 8:2:5, A’s share was \(8/15\) and B’s share was \(2/15\). Under the new ratio 3:2, A’s share becomes \(3/5=9/15\) and B’s share becomes \(2/5=6/15\). Thus, A gains \(1/15\) and B gains \(4/15\), so the gaining ratio is \(1:4\). Remember that 3:2 is the new profit-sharing ratio, not the gaining ratio.

A’s old share was \(5/(5+8+2)=5/15\). After B’s retirement, A’s new share is \(4/(4+1)=4/5=12/15\). Therefore, A’s gaining share = new share − old share = \(12/15-5/15=7/15\). Option B represents A’s new share, not the gain. Exam tip: Always calculate gaining share as the new share minus the old share.

C’s old profit share was 2/(5+8+2) = 2/15. Under the new ratio 4:1, C’s new share is 1/(4+1) = 1/5 = 3/15. Therefore, C’s gaining share is 3/15 − 2/15 = 1/15. The fraction 1/5 is C’s new share, not the gain. In the exam, calculate the old and new shares separately and then subtract the old share from the new share.

Under the old ratio, A, B and C had shares of \(5/15\), \(8/15\) and \(2/15\), respectively. Under the new ratio of 4:1, A gets \(4/5=12/15\) and C gets \(1/5=3/15\). Therefore, A’s gain is \(12/15-5/15=7/15\), while C’s gain is \(3/15-2/15=1/15\). Hence, the gaining ratio is \(7/15:1/15=7:1\). Remember: the new ratio is 4:1, but the gaining ratio is calculated from the increase in each continuing partner’s share.

In the new ratio 12:11, A’s new share is \(12/(12+11)=12/23\). Gaining share = new share − old share = \(12/23-1/2=24/46-23/46=1/46\). Therefore, option A is correct. Option B is only A’s new share, while option C is A’s old share. Exam tip: Always subtract the old share from the new share to find the gaining share.

A gaining share is the increase in a continuing partner’s share after retirement. Hence it is \(\text{New Share} - \text{Old Share}\). A negative difference indicates sacrifice, not gain. Exam tip: always check the sign of the difference.

After D retires, the ratio among A, B and C remains 5:3:2 because the question states that they continue in their old ratio. D’s share of 1/11 is effectively absorbed by the continuing partners in that same proportion, so the new ratio is 5:3:2. Exam tip: When the continuing partners retain their old mutual ratio, that old ratio becomes the new ratio after retirement.

C’s profit share is 4 parts, and A, B and D acquire it in the ratio 2:1:1. Since the total of the acquisition ratio is also 4, A, B and D receive 2, 1 and 1 additional parts respectively. Thus, A = 6 + 2 = 8, B = 5 + 1 = 6 and D = 3 + 1 = 4. Therefore, the new ratio is 8:6:4. Exam tip: Distribute the retiring partner’s share among the gaining partners according to the specified acquisition ratio, then add those gains to their old shares.

B’s original share is 5 out of the total 18 parts. This share is acquired by A, C and D in the ratio 3:1:1; hence, A receives 3 parts, C receives 1 part and D receives 1 part out of B’s 5 parts. Therefore, the new shares are A = 7 + 3 = 10, C = 4 + 1 = 5 and D = 2 + 1 = 3. Thus, the new ratio is 10:5:3. Exam tip: Divide the outgoing partner’s share according to the gaining ratio and add each gain to the partner’s old share.

A’s original profit share is 4 parts. Since B, C and D acquire it in the ratio 2:0:2, B receives 2 parts, C receives no additional share, and D receives 2 parts. Therefore, the new shares are B = 5 + 2 = 7, C = 6 + 0 = 6, and D = 3 + 2 = 5. Hence, the new ratio is 7:6:5. Exam tip: Apply the acquisition ratio to the retiring partner’s share, not directly to the old total ratio.