New Ratios

D’s old profit-sharing fraction is \(2/15\). Dividing it among A, B and C in the ratio 1:2:1 gives them additional shares of \(1/30\), \(1/15\) and \(1/30\), respectively. Thus, the new shares are A = \(6/15+1/30=13/30\), B = \(4/15+1/15=10/30\), and C = \(3/15+1/30=7/30\). Therefore, the new ratio is 13:10:7. Exam tip: Divide the retiring partner’s share in the stated gaining ratio and add each gain to the partner’s old share.

A’s profit share is 9/16. Since B, C and D acquire it equally, each receives 3/16. Therefore, B’s new share is 4/16 + 3/16 = 7/16, C’s is 2/16 + 3/16 = 5/16, and D’s is 1/16 + 3/16 = 4/16. Hence, the new ratio is 7:5:4. Exam tip: Divide the retiring partner’s share equally among the gaining partners and add each gain to their old share.

After retirement or death the whole future profit must be shared by remaining partners only. So new shares must add to one.

The sum of the terms of the new ratio is 44 + 31 = 75. Hence, the first partner’s new share is 44/75 and the second partner’s new share is 31/75. Option B incorrectly treats the ratio terms as fractions of each other, while option C reverses the partners’ shares. Exam tip: To convert a ratio into individual shares, divide each ratio term by the sum of all ratio terms.

Both shares have the same denominator, 50. Thus, \(\frac{28}{50}:\frac{17}{50}\) simplifies by cancelling the common denominator to \(28:17\). \(17:28\) reverses the order of the partners given in the question. Exam tip: when two fractions have the same denominator, their ratio is the ratio of their numerators.

Goodwill burden is for the extra share received by remaining partners. Therefore gaining ratio is the correct basis.

The governing rule is that each continuing partner’s new share equals the old share plus the portion acquired from the retiring partner. The original ratio totals 4 + 3 + 5 = 12. Thus A’s old share is 4/12, B’s old share is 3/12, and C’s retiring share is 5/12. A and B acquire C’s share in the ratio 2:3. A receives 2/5 × 5/12 = 2/12, while B receives 3/5 × 5/12 = 3/12. Therefore A’s new share is 4/12 + 2/12 = 6/12, and B’s new share is 3/12 + 3/12 = 6/12. Their ratio is 6:6, which simplifies to 1:1. Option A is correct; the other options represent an old or acquisition ratio or an incorrect calculation.

The governing concept is that the gaining ratio is calculated from the increase in each continuing partner’s share, not from the final new ratio alone. The original total is 10, so A’s old share is 1/10 and B’s old share is 3/10. Under the new ratio 5:5, each receives 5/10. A’s gain is therefore 5/10−1/10=4/10, while B’s gain is 5/10−3/10=2/10. The gains are 4/10:2/10, which reduce to 2:1. Option A is correct. Option B incorrectly assumes equal gains because the final shares are equal; C gives the new ratio rather than the gains; and D is merely the old ratio of A and B.

The governing principle is to compare each continuing partner’s old share with the corresponding new share. The original ratio totals 10, giving A 5/10 and B 4/10. The new ratio 6:4 already has a total of 10, so A’s new share is 6/10 and B’s new share is 4/10. A has gained 6/10−5/10=1/10. B has gained 4/10−4/10=0. Therefore, the gaining ratio is 1/10:0, or 1:0. This means A alone gains C’s share, while B’s share remains unchanged. Option A is correct; B reverses the result, C wrongly assumes equal gains, and D states the new ratio rather than the gaining ratio.

The governing rule is to add the retiring partner’s share, divided according to the stated acquisition ratio, to each continuing partner’s old share. The original total is 10. A’s share is 4/10, and B, C and D each have 2/10. Since A’s 4/10 share is acquired equally by three partners, each receives 4/30. Each continuing partner therefore has 2/10+4/30 = 6/30+4/30 = 10/30 = 1/3. Their shares are equal, so the new ratio is 1:1:1. Option A is correct. Options B and C are not reduced or correctly adjusted final ratios, while D gives an unsupported ratio and does not result from the calculation.

The governing concept is the gaining ratio: a continuing partner’s gain equals the new profit share minus the old profit share. The old ratio is 2:7:6, whose total is 15, so A’s old share is 2/15. After B retires, A and C divide future profits in the ratio 3:5, whose total is 8; hence A’s new share is 3/8. Therefore, A’s gain is 3/8 − 2/15. Using 120 as the common denominator, 3/8 = 45/120 and 2/15 = 16/120; the difference is 29/120. Thus option C is correct. The other options do not equal this properly converted difference between A’s new and old shares.

The governing rule for a gaining ratio is gain = new share − old share. The old ratio 2:5:3 totals 10, so A’s old share is 2/10 and C’s old share is 3/10. The new ratio 7:5 totals 12, so A’s new share is 7/12 and C’s new share is 5/12. A’s gain is 7/12 − 2/10 = 35/60 − 12/60 = 23/60. C’s gain is 5/12 − 3/10 = 25/60 − 18/60 = 7/60. Removing the common denominator gives the gaining ratio 23:7, so option A is correct. Option C is merely the new profit-sharing ratio, not the ratio in which the continuing partners gain the retiring partner’s share. Option B reverses the two gains, and option D uses neither correct calculation.