New Ratios

If no information is given the old mutual ratio of continuing partners becomes the new ratio. Therefore A and B new ratio is 5:4.

After removing B the old shares of A C and D are 2:4:1. If no information is given this becomes the new ratio.

The retiring or deceased partner share is added to continuing partners. So new share is the sum of old share and gained share.

Gaining ratio shows the ratio in which continuing partners increased their profit share. It is very useful in goodwill compensation.

The total ratio is 7+5+3=15, so C’s share is 3/15. A acquires 2/3 of C’s share, which is 2/15, while B acquires 1/3, which is 1/15. Therefore, A’s new share is 7/15+2/15=9/15 and B’s is 5/15+1/15=6/15. Thus, the new ratio is 9:6, simplified to 3:2. Exam tip: Add each partner’s acquired portion of the retiring partner’s share to that partner’s old share.

C’s original share was 5/10. When C retires, A acquires 1/5 of C’s share, which is 1/10, while B acquires 4/5 of C’s share, which is 4/10. Thus, A’s new share is 3/10 + 1/10 = 4/10, and B’s new share is 2/10 + 4/10 = 6/10. Simplifying 4:6 gives the new ratio 2:3. Exam tip: 1:4 is the gaining ratio, not the new profit-sharing ratio.

B’s original share is 3/9. A acquires 3/4 of it, which is 1/4, while C acquires 1/4 of it, which is 1/12. Therefore, A’s new share is 4/9 + 1/4 = 25/36 and C’s new share is 2/9 + 1/12 = 11/36. Hence, the new ratio is 25:11. Exam tip: Apply the acquisition ratio only to the deceased partner’s share, not to the partners’ original shares.

The correct answer is Option A, 9:11. The old ratio A:B:C is 1:4:5, so the old shares are 1/10, 4/10 and 5/10. A dies and gives up 1/10. B and C acquire this share equally, so each receives 1/20. B's new share is 4/10 + 1/20 = 8/20 + 1/20 = 9/20. C's new share is 5/10 + 1/20 = 10/20 + 1/20 = 11/20. Therefore B:C = 9:11. Option A is correct. Option B, 4:5, simply repeats the old ratio of B and C and ignores the share acquired from A. Option C, 1:1, would mean B and C have equal total shares, but C originally had a larger share and remains larger even after equal acquisition. Option D, 5:6, does not follow the calculation and gives an incorrect difference between the new shares. Notice that equal acquisition means equal additions, not equal final shares. Memory cue: add the deceased partner's surrendered share equally to each continuing partner, then compare the totals.

The governing concept is that the gaining ratio is calculated by comparing each continuing partner’s new share with the corresponding old share. The old ratio totals 10, so A’s old share is 6/10 and B’s old share is 1/10. The new ratio totals 7, so A’s new share is 5/7 and B’s new share is 2/7. A’s gain is 5/7 − 6/10 = 50/70 − 42/70 = 8/70. B’s gain is 2/7 − 1/10 = 20/70 − 7/70 = 13/70. Since the common denominator is irrelevant when forming the ratio, the gaining ratio is 8:13. Option B is merely the new ratio, option C is the old ratio, and option D incorrectly assumes equal gains. Therefore option A is correct.

To compare gains, first express old and new shares as fractions of the whole. The original total is 10, so A held 3/10 and B held 2/10. The new ratio 2:3 has total 5, so A now holds 2/5=4/10 and B holds 3/5=6/10. A therefore gains 4/10−3/10=1/10. B gains 6/10−2/10=4/10. Since 4/10 is greater than 1/10, B acquires the larger share. Thus option A is correct. Option B reverses the comparison, option C ignores the unequal gains, and option D is impossible because both continuing partners together acquire C’s share.

Gaining ratio is the same ratio in which continuing partners take the retiring partner share. Hence it will be 7:5.

When share is acquired in old mutual ratio the mutual relation of continuing partners does not change. So new ratio remains the same.

The greatest common divisor of the terms 10 and 6 is 2. Dividing both terms by 2 gives 10:6 = 5:3, so the simplified new ratio is 5:3. The ratio 10:6 is not in lowest terms, while 3:5 reverses the order of the partners. Exam tip: always divide the terms by their greatest common divisor before recording a final ratio.

After removing B the old mutual ratio of A C and D is 5:4:1. If no information is given this is the new ratio.

A’s old share was 3/(3+1+6) = 3/10. After C’s retirement, the new ratio of A and B is 2:1, so A’s new share is 2/3. Therefore, A’s gain = new share − old share = 2/3 − 3/10 = 20/30 − 9/30 = 11/30. The value 3/10 is A’s old share, not the gain. Exam tip: A partner’s gaining share is calculated as new share minus old share.

B’s old profit share was \(1/(3+1+6)=1/10\). After C’s retirement, B’s new share in the 2:1 ratio is \(1/(2+1)=1/3\). Therefore, B’s gaining share = new share − old share = \(1/3-1/10=10/30-3/30=7/30\). Exam tip: for a gaining partner, subtract the old share from the new share.

After retirement or death only continuing partners run the firm. Therefore new ratio is required for future profits and losses.

Gaining ratio shows who got extra profit share. Therefore goodwill compensation is made from continuing partners in this ratio.

When new ratio is clearly given future profits and losses are shared in that ratio. Gaining ratio is separately used to find gains.

D share is 1/10 and in 2:1:2 A gets 1/25 B gets 1/50 and C gets 1/25. New shares are 17/50 21/50 and 12/50.

A gain is 4/7 - 5/10 = 1/14 and B gain is 3/7 - 3/10 = 9/70. With common denominator the ratio is 5:9 not 1:11.

Under the old ratio, A's share was 5/10 and B's share was 3/10. Under the new ratio, A's share is 4/7 and B's share is 3/7. Therefore, A's gain = 4/7 − 5/10 = 5/70, while B's gain = 3/7 − 3/10 = 9/70. Hence, the gaining ratio is 5:9. Option 4:3 is the new profit-sharing ratio, not the gaining ratio. Exam tip: Calculate each partner's gain as new share − old share, then compare the gains.

Before retirement, both A and B had a share of 4/10. Under the new ratio, A’s share becomes 7/12, so A’s gain is 7/12 − 4/10 = 11/60. B’s share becomes 5/12, so B’s gain is 5/12 − 4/10 = 1/60. Therefore, A gained more than B. Exam tip: Compare the gain of each continuing partner by subtracting the old share from the new share.

Under the old ratio 4:4:2, A's share was 4/10 and B's share was 4/10. Under the new ratio 7:5, A's share is 7/12 and B's share is 5/12. Thus, A's gain is 7/12 − 4/10 = 11/60, while B's gain is 5/12 − 4/10 = 1/60. Therefore, the gaining ratio is 11/60:1/60 = 11:1. Exam tip: Gaining ratio = new share − old share; the retired partner's share is not included in the gaining ratio.

D’s original share is 1/10. On distributing it in the ratio 3:2:1, A, B and C gain 1/20, 1/30 and 1/60 respectively. Their original shares are 4/10, 3/10 and 2/10, or 24/60, 18/60 and 12/60. Adding the gains gives 27/60, 20/60 and 13/60, so the new ratio is 27:20:13. The ratio 4:3:2 merely retains the old ratio among A, B and C without adding D’s redistributed share. Exam tip: first calculate the retiring partner’s share, then distribute it according to the gaining ratio.