New Ratios

Since the old ratio is 4:4:2, both A and B originally held \(4/10=2/5\). Under the new ratio, A’s share is \(7/12\) and B’s share is \(5/12\). Thus, A’s gain is \(7/12-2/5=35/60-24/60=11/60\), whereas B’s gain is \(5/12-2/5=25/60-24/60=1/60\). Therefore, A gains the greater share. Exam tip: Gain = new share − old share.

Take the total profit as 10 parts. The old shares of A, B and C are 3/10, 2/10 and 5/10, respectively. A and B acquire C’s 5/10 share in the ratio 4:1, so A receives 4/10 and B receives 1/10. Therefore, A’s new share is 3/10 + 4/10 = 7/10, while B’s new share is 2/10 + 1/10 = 3/10. Hence, the new ratio is 7:3. Exam tip: Divide the retiring partner’s share according to the acquisition ratio and add those portions to the partners’ old shares.

The direct answer is A, 13:7. The old total is 1+6+3=10, so A has 1/10, B has 6/10, and C has 3/10. A retires and gives the whole 1/10 share to B and C equally. Each receives half of 1/10, which is 1/20. B’s new share is 6/10+1/20=12/20+1/20=13/20. C’s new share is 3/10+1/20=6/20+1/20=7/20. Hence the new ratio is 13:7. Option A is correct. Option B, 6:3, ignores the extra share. Option C, 1:1, describes equal acquisition, not final shares. Option D, 7:4, does not result from the calculation. Always distinguish the ratio in which a retiring share is acquired from the final profit-sharing ratio.

A’s old share was 4/10. Under the new ratio of 3:2, A’s new share is 3/5, or 6/10. Therefore, A’s gaining ratio = new share − old share = 6/10 − 4/10 = 2/10 = 1/5. Option C represents A’s old share, not the gain. Exam tip: On a partner’s retirement, calculate the gaining ratio as the new share minus the old share.

B’s old share was \(2/10=1/5\). After C’s retirement, B’s new share under the ratio 3:2 is \(2/5=4/10\). Therefore, B’s gaining share = new share − old share = \(4/10-2/10=2/10=1/5\). Option 2/5 is B’s new share, not the gain. Exam tip: Always subtract the old share from the new share to calculate a partner’s gaining share.

Under the old ratio, A’s share was 4/10 = 2/5 and B’s share was 2/10 = 1/5. Under the new ratio, A receives 3/5 and B receives 2/5. Thus, A’s gain is 3/5 − 2/5 = 1/5, while B’s gain is 2/5 − 1/5 = 1/5. Therefore, the gaining ratio is 1/5 : 1/5 = 1:1. Exam tip: Calculate each continuing partner’s gain by subtracting the old share from the new share.

D share is 2/10. In 2:1:1 A gets 1/10 B gets 1/20 and C gets 1/20 so new shares are 6/10 3/20 and 5/20 which simplify to 12:3:5.

The total ratio is 10, so D’s share is 2/10. A, B and C acquire this share in the ratio 2:1:1; therefore, their gains are 1/10, 1/20 and 1/20 respectively. Their new shares are A = 5/10 + 1/10 = 6/10, B = 1/10 + 1/20 = 3/20 and C = 2/10 + 1/20 = 5/20. Converting them to a common denominator gives 12:3:5. Exam tip: on retirement, distribute the retiring partner’s share according to the gaining ratio, not the old profit-sharing ratio.

B’s old share was 5/10. Since B’s share is acquired in the ratio 3:2, A receives 3/5 × 5/10 = 3/10 and C receives 2/5 × 5/10 = 2/10. Therefore, A’s new share is 2/10 + 3/10 = 5/10, while C’s new share is 3/10 + 2/10 = 5/10. Hence, the new ratio is 5:5, or 1:1. The ratio 3:2 represents only the acquisition of B’s share, not the final ratio. Exam tip: New share = old share + acquired share.

A’s old share was \(\frac{1}{1+3+6}=\frac{1}{10}\). Under the new ratio 2:3, A’s new share is \(\frac{2}{5}=\frac{4}{10}\). Therefore, A’s gain is \(\frac{4}{10}-\frac{1}{10}=\frac{3}{10}\). The fraction \(\frac{2}{5}\) is A’s new share, not the gain. Exam tip: gain = new share − old share.

B's old share was 3/(1+3+6) = 3/10, while his new share is 3/(2+3) = 3/5 = 6/10. Therefore, B's gain = new share − old share = 6/10 − 3/10 = 3/10. Option B represents B's new share, not his gain. Exam tip: calculate gain by subtracting the old share from the new share.

Under the old ratio, A’s share was 1/10 and B’s share was 3/10. Under the new ratio 2:3, A’s share becomes 2/5, or 4/10, and B’s share becomes 3/5, or 6/10. Thus, A’s gain is 4/10 − 1/10 = 3/10, while B’s gain is 6/10 − 3/10 = 3/10. Therefore, the gaining ratio is 3/10 : 3/10 = 1:1. Exam tip: calculate each continuing partner’s new share minus old share; do not use the new ratio directly as the gaining ratio.

The original total ratio is 10, so D’s share is 5/10. When this share is acquired in the ratio 1:2:2, A receives 1/10, B receives 2/10, and C receives 2/10. Therefore, the new shares are A = 2/10 + 1/10 = 3/10, B = 2/10 + 2/10 = 4/10, and C = 1/10 + 2/10 = 3/10. Hence, the new ratio is 3:4:3. Exam tip: Divide the deceased partner’s share according to the gaining ratio and add each gain to the partner’s original share.

A’s old share was 3/(3+5+2) = 3/10. After C’s retirement, A’s new share is 1/(1+2) = 1/3. Therefore, A’s gain is 1/3 − 3/10 = 1/30, so A gained 1/30 of the profit. Option B reverses the direction of the change and incorrectly calls the gain a sacrifice. Exam tip: calculate new share − old share; a positive result indicates a gain.

B’s old profit share was \(\frac{5}{10}=\frac{1}{2}\). Under the new ratio 1:2, B’s new share is \(\frac{2}{3}\). Therefore, B’s gain = new share − old share = \(\frac{2}{3}-\frac{1}{2}=\frac{4-3}{6}=\frac{1}{6}\). Option C represents B’s old share, not the gain. Exam tip: calculate a partner’s gain by subtracting the old share from the new share.

Before C’s retirement, A’s share was 3/10 and B’s share was 5/10. Under the new ratio 1:2, A’s share becomes 1/3 and B’s share becomes 2/3. Therefore, A’s gain is 1/3 − 3/10 = 1/30, while B’s gain is 2/3 − 5/10 = 1/6 = 5/30. Hence, the gaining ratio is 1/30 : 5/30 = 1:5. Option 1:2 is the new profit-sharing ratio, not the gaining ratio. Exam tip: Calculate each partner’s gain as new share minus old share, then simplify the ratio of gains.

D share is 4/10. A gets 3/10 B gets 1/10 and C gets nothing so new shares are 7/10 2/10 and 1/10.

B’s old share was 4/10 under the ratio 5:4:1. In the new ratio 8:2, B’s share is 2/10. Therefore, B’s sacrifice = old share − new share = 4/10 − 2/10 = 2/10. Hence, B sacrificed 2/10 share. Exam tip: Compare a partner’s old and new shares—if the old share is larger, there is a sacrifice; if the new share is larger, there is a gain.

A’s old share was 5 out of 10, or 5/10. After C’s retirement, the new ratio of A and B is 8:2, so A’s new share is 8/10. Therefore, A’s gain = New share − Old share = 8/10 − 5/10 = 3/10. Exam tip: In a partner’s retirement, calculate gain by subtracting the old share from the new share.

Gaining ratio is made from positive increase. A decreased share is sacrifice so it is not added to gain.

Direct answer: Option A, 3:1. The old ratio of A, B and C is 4:3:3, so B’s old share is 3/10 and C’s old share is 3/10. After A retires, the new ratio of B and C is 6:4, or shares 6/10 and 4/10. Gain means new share minus old share. B gains 6/10−3/10=3/10. C gains 4/10−3/10=1/10. Their common denominator is 10, so the gaining ratio is 3:1. Option A is correct. Option B, 6:4, is the new ratio, not the gain ratio. Option C, 1:1, would wrongly assume equal gains, although B gains three times as much as C. Option D, 3:3, also states equal gains and is therefore wrong. A useful check is that total gain is 3/10+1/10=4/10, exactly A’s old share. Memory cue: gaining ratio = new share − old share.

B’s old share was 4/10. Since A, C and D acquire it in the ratio 1:2:1, C receives 2/4 of B’s share: 2/4 × 4/10 = 2/10. Therefore, C’s new share is 3/10 + 2/10 = 5/10 = 1/2. Exam tip: Add the partner’s old share to the share acquired from the retiring partner; do not report only the acquired portion.

The total of the original ratio is 10, so B’s share is 4/10. Distributed in the ratio 1:2:1, this gives A, C and D gains of 1/10, 2/10 and 1/10 respectively. Their new shares are therefore 2/10+1/10=3/10, 3/10+2/10=5/10 and 1/10+1/10=2/10. Hence, the new ratio is 3:5:2. Exam tip: Divide the retiring partner’s share according to the gaining ratio and add each gain to the partner’s old share.

A moved from 3/10 to 6/10 and B from 2/10 to 4/10. Gains are 3/10 and 2/10 so they are not equal.

Under the old ratio, A’s share was 3/10 and B’s share was 2/10. Expressing the new ratio 6:4 on a base of 10 gives A’s share as 6/10 and B’s share as 4/10. Thus, A’s gain = 6/10 − 3/10 = 3/10, and B’s gain = 4/10 − 2/10 = 2/10. Therefore, the gaining ratio is 3:2. Option C is the new profit-sharing ratio, not the gaining ratio. Exam tip: Gaining ratio = New ratio − Old ratio.