New Ratios

D share of 2/10 is divided in 1:2:1 so A gets 1/20 B gets 1/10 and C gets 1/20. New shares are 7/20 8/20 5/20 so the ratio is 7:8:5.

C share of 3/15 is acquired in 4:1. New shares of A and B are 47/75 and 28/75 so the ratio is 47:28.

B’s original share is 3 parts. A and C acquire this share in the ratio 1:2, so A receives 1 part and C receives 2 parts. Therefore, A’s new share is 4+1=5 and C’s new share is 5+2=7. Hence, the new ratio is 5:7. The ratio 1:2 represents only the manner of acquiring B’s share, not the new overall ratio. Exam tip: Add each continuing partner’s acquired share to their original share after removing the retiring or deceased partner.

A’s share is \(8/15\) of the total profit. B receives \(3/8\) of A’s share, i.e. \(3/15\), while C receives \(5/15\). Thus, B’s new share is \(5/15+3/15=8/15\), and C’s new share is \(2/15+5/15=7/15\). Therefore, the new ratio is \(8:7\). The ratio 5:2 is the old ratio, while 3:5 is only the gaining ratio for A’s share. Exam tip: First calculate the retiring partner’s share, then distribute it according to the gaining ratio.

A’s old share was 6/(6+4+5) = 6/15 = 2/5. After C’s retirement, A’s new share is 5/(5+4) = 5/9. Therefore, A’s gaining share = new share − old share = 5/9 − 2/5 = (25−18)/45 = 7/45. Exam tip: Calculate the gaining share as the increase in the partner’s share, not as the new share alone.

B’s old profit-sharing share was \(\frac{4}{6+4+5}=\frac{4}{15}\). After C’s retirement, B’s new share is \(\frac{4}{5+4}=\frac{4}{9}\). Therefore, B’s gaining share = new share − old share = \(\frac{4}{9}-\frac{4}{15}=\frac{8}{45}\). In such questions, subtract the partner’s old share from the new share to find the gaining share.

The old ratio is 6:4:5, so A, B and C had old shares of 18/45, 12/45 and 15/45 respectively. In the new ratio 5:4, A and B receive 25/45 and 20/45 respectively. Therefore, A’s gain is 25/45 − 18/45 = 7/45, while B’s gain is 20/45 − 12/45 = 8/45. Hence, the gaining ratio is 7:8. Remember that gaining ratio is calculated as new share minus old share.

The correct method is to compare each continuing partner’s old share with the new share. The old total is 4+6+5=15, so A’s old share is 4/15 and B’s old share is 6/15=2/5. The new ratio 2:5 has total 7, so A’s new share is 2/7 and B’s new share is 5/7. For A, 2/7−4/15=(30−28)/105=2/105, which is an increase. For B, 5/7−6/15=5/7−2/5=(25−14)/35=11/35, also an increase. Therefore neither partner’s share has decreased, so option D is correct. Option A is a tempting error caused by comparing numerators without equal denominators.

A’s old profit share is \(\frac{4}{4+6+5}=\frac{4}{15}\). After C’s retirement, A’s new share is \(\frac{2}{2+5}=\frac{2}{7}\). Therefore, A’s gaining share = new share − old share = \(\frac{2}{7}-\frac{4}{15}=\frac{30-28}{105}=\frac{2}{105}\). In such questions, subtract the old share from the new share to find the gaining share.

B’s old profit share was 6 out of the total ratio 4+6+5=15, so it was 6/15=2/5. After C’s retirement, B’s new share is 5/(2+5)=5/7. Therefore, B’s gaining share = new share − old share = 5/7 − 2/5 = 25/35 − 14/35 = 11/35. The option 5/7 represents B’s new share, not the gaining share. Exam tip: Always calculate the gaining share as the difference between the new share and the old share.

From the old ratio 2:7:6, A’s old share is 2/15 and B’s old share is 7/15. Under the new ratio 3:4, A’s new share is 3/7 and B’s new share is 4/7. Therefore, A’s gain = 3/7 − 2/15 = 31/105, while B’s gain = 4/7 − 7/15 = 11/105. Hence, the gaining ratio is 31:11. The ratio 3:4 is the new profit-sharing ratio, not the gaining ratio. Exam tip: Calculate each continuing partner’s gain as new share minus old share.

Initially, A, B and C have shares of 5, 8 and 7 parts, respectively. B’s 8 parts are acquired by A and C in the ratio 5:3, so A receives 5 additional parts and C receives 3 additional parts. Thus, A’s new share is 5+5=10 and C’s new share is 7+3=10. Therefore, the new ratio is 10:10, or 1:1. Exam tip: Divide the retiring partner’s share in the gain ratio of the acquiring partners and add those gains to their old shares.

In the old ratio, A’s share is 9 parts. B and C acquire these 9 parts in the ratio 2:7, so B receives 2 parts and C receives 7 parts. Thus, B’s new share is 5+2=7 and C’s new share is 4+7=11, giving the new ratio 7:11. Exam tip: Do not treat the gaining ratio as the new ratio; add the acquired portions to the partners’ original shares.

When remaining partners continue in old ratio the retiring share is also treated as acquired in that ratio.

After B retires, only A and C remain as partners. Since they continue to share profits in their old mutual ratio, the new ratio of A:C remains equal to their original shares, 11:5. In 4:5, the share of B has been incorrectly used for A. Exam tip: When the remaining partners continue in their old mutual ratio, retain only the ratio terms belonging to those partners who continue.

After A’s death, the old mutual ratio of B and C remains 6:9. Dividing both terms by their highest common factor, 3, gives 6:9 = 2:3. Therefore, the new profit-sharing ratio is 2:3. Exam tip: When partners continue in their old mutual ratio, retain their original relative shares and reduce the ratio to its simplest form. Option 6:9 is not simplified, while 3:2 reverses the partners’ order.

A’s old profit share was 4 out of a total ratio of 20, so it was 4/20 = 1/5. After C’s retirement, A’s new share is 3/(3+5) = 3/8. Therefore, A’s gaining share = new share − old share = 3/8 − 1/5 = 15/40 − 8/40 = 7/40. Option B is A’s new share, not the gaining share. Exam tip: Always calculate the gaining share as the new share minus the old share.

B’s old profit share was 9 out of the total 4+9+7=20, so it was 9/20. Under the new ratio 3:5, B’s new share is 5/(3+5)=5/8. Therefore, B’s gaining share = new share − old share = 5/8 − 9/20 = 25/40 − 18/40 = 7/40. Remember that the gaining share is the increase in share, not the partner’s new share itself; hence 5/8 is not the answer.

Under the old ratio, A’s share was \(4/20=8/40\) and B’s share was \(9/20=18/40\). Under the new ratio of 3:5, A’s share is \(3/8=15/40\) and B’s share is \(5/8=25/40\). Thus, A gains \(15/40-8/40=7/40\), while B gains \(25/40-18/40=7/40\). Since both partners gain equally, the gaining ratio is 1:1. Exam tip: Calculate each partner’s gain as new share minus old share, then simplify the ratio.

The total of the old ratio is 15. A’s old share was 8/15, whereas the new share is 1/2 = 7.5/15; therefore, A’s share decreases by 1/30. B’s share increases from 3/15 to 1/2, so it has not decreased. Exam tip: After a partner’s retirement, compare each continuing partner’s old share with the new share.

The total old ratio is 8+3+4=15, so B’s old share was 3/15=1/5. Under the new 1:1 ratio, B’s share becomes 1/2. Therefore, B’s gaining share = new share − old share = 1/2 − 1/5 = 3/10. Exam tip: a partner’s gaining share is calculated by subtracting the old share from the new share; 1/2 is B’s new share, not the gain.

The gaining ratio must be calculated from the increase in each continuing partner’s share. The old total is 15, so A’s old share is 8/15 and B’s old share is 3/15=1/5. Under the new 1:1 ratio, each receives 1/2. A’s gain is 1/2−8/15=(15−16)/30, which is actually −1/30; A’s share decreases. B’s gain is 1/2−1/5=3/10=9/30. Since A is not a gainer and B gains 9/30, the gaining ratio is 0:9, or 0:1 after simplification. Therefore option B is correct. Options C and D wrongly treat A as a gainer or assume equal gains, while 7:9 confuses old-share comparisons with gains.

B’s old profit share is 2/10. When this is acquired in the ratio 4:1, A receives 2/10 × 4/5 = 8/50 and C receives 2/10 × 1/5 = 2/50. Thus, A’s new share is 5/10 + 8/50 = 33/50, while C’s new share is 3/10 + 2/50 = 17/50. Therefore, the new ratio is 33:17. Exam tip: First write the retiring partner’s old share, then distribute it according to the acquiring partners’ ratio.

C’s original share is \(2/15\). Since A and B acquire it in the ratio 1:4, A gets \(2/75\) and B gets \(8/75\). A’s new share is \(10/15+2/75=52/75\), while B’s new share is \(3/15+8/75=23/75\). Therefore, the new ratio is \(52:23\). Exam tip: Divide the retiring partner’s share in the acquisition ratio and add the respective gains to the old shares.

A’s original share is 3 parts. B and C acquire this share in the ratio 2:1, so B receives 2 parts and C receives 1 part. Thus, B’s new share is 8+2=10 and C’s new share is 4+1=5. The new ratio is therefore 10:5, which reduces to 2:1. The option 5:2 is incorrect because it does not correctly add the acquired portions to B’s and C’s original shares. Exam tip: calculate each partner’s gain first, add it to the old share, and then simplify the resulting ratio.