New Ratios

The governing concept is the gaining ratio on retirement. It is calculated by comparing each continuing partner’s new share with his or her old share. A’s old share is 4/9 and new share is 5/9, so A gains 5/9 − 4/9 = 1/9. B’s old share is 3/9 and new share is 4/9, so B gains 4/9 − 3/9 = 1/9. Therefore, the gains are 1/9 : 1/9, which simplify to 1:1. Option A is correct. The new profit-sharing ratio itself, 5:4, is not the gaining ratio, and the old ratio 4:3 does not measure the increase received by the continuing partners.

Under the stated assumption, A and B retain their old relative shares of 3 and 2. Therefore, their new ratio is 3:2.

A and C acquire B’s share in the same relative ratio as their old shares, 5:2. Adding B’s share in that proportion does not change their relative ratio, so the new ratio is 5:2.

The total old shares are 9, so C’s share is 2/9. Each of A and B acquires 1/9. Their new shares are A: 4/9 + 1/9 = 5/9 and B: 3/9 + 1/9 = 4/9. Thus, the new ratio is 5:4.

C’s share is 2/9. A and B each acquire half of it, or 1/9. Their new shares are 4/9 + 1/9 = 5/9 for A and 3/9 + 1/9 = 4/9 for B, so the new ratio is 5:4.

The total old shares are 16, so C’s share is 4/16. Dividing it in the ratio 3:1 gives A an additional 3/16 and B an additional 1/16. Their new shares are 10/16 and 6/16, which simplify to 5:3.

The total old shares are 10, so C’s share is 5/10. Dividing it in the ratio 1:4 gives A an additional 1/10 and B an additional 4/10. Their new shares are 2/10 + 1/10 = 3/10 and 3/10 + 4/10 = 7/10, so the new ratio is 3:7.

C’s share is 5/10. When it is acquired by A and B in the ratio 1:4, A receives 1/10 and B receives 4/10. Thus, A’s new share is 2/10 + 1/10 = 3/10, while B’s new share is 3/10 + 4/10 = 7/10. Therefore, the new ratio is 3:7. Exam tip: Divide the retiring partner’s share according to the gaining ratio and add each gain to the partner’s old share.

The correct answer is Option A, 2:3. A, B and C originally share profits in 1:2:3. When A retires, only B and C remain. No information is given that they will change their relative claims, so the standard rule is that continuing partners retain their old mutual ratio. B's old share is 2 parts and C's old share is 3 parts; removing A's 1 part leaves B:C = 2:3. Option A is correct. Option B, 1:2, incorrectly uses A's old share and B's old share rather than the remaining partners' shares. Option C, 1:3, omits B's correct two-part share and is unsupported. Option D, 3:2, reverses the two remaining partners' proportions. The total old ratio was six parts, but the new ratio need not be written with a total of six; the relative shares of B and C are what matter, and 2:3 already has the correct relationship. Memory cue: on retirement, if no new arrangement is stated, cross out the retiring partner and retain the remaining partners' old ratio.

The governing concept is the new profit-sharing ratio after retirement. The old total is 3+3+2=8, so A’s retiring share is 3/8. B and C acquire this share equally; therefore each receives 3/16. B’s original share is 3/8=6/16, so B’s new share becomes 6/16+3/16=9/16. C’s original share is 2/8=4/16, so C’s new share becomes 4/16+3/16=7/16. Removing the common denominator gives the new ratio 9:7, so option A is correct. Equal acquisition refers only to the additional share received, not to the final shares. Thus options B, C and D fail to add A’s share correctly.

The total initial share is 12, so B’s share is 4/12. Since A and C acquire it in the ratio 2:1, A receives 8/36 and C receives 4/36. Therefore, A’s new share is 15/36 + 8/36 = 23/36, while C’s new share is 9/36 + 4/36 = 13/36. Hence, the new ratio is 23:13. Exam tip: Divide the retiring partner’s share according to the acquisition ratio and add each gain to the partner’s old share.

A’s share is \(2/10\). B and C acquire it in the ratio 3:2, so B receives \((3/5)\times(2/10)=3/25\) and C receives \((2/5)\times(2/10)=2/25\). Therefore, B’s new share is \(5/10+3/25=31/50\), while C’s new share is \(3/10+2/25=19/50\). Hence, the new ratio is 31:19. Option 8:5 is only B and C’s old ratio and does not account for the transfer of A’s share. Exam tip: First calculate the retiring partner’s share, then distribute it according to the stated acquisition ratio.

A continuing partner’s new share equals their old share plus the share they gain from the retiring partner. The gained share is added because it increases the continuing partner’s share.

The gaining ratio shows the relative shares in which the continuing partners gain the retiring partner’s share. It is used when adjusting goodwill on retirement.

Subtract A’s old share from the new share: 7/12 − 3/5 = 35/60 − 36/60 = −1/60. The negative result means A’s share decreased, so A sacrificed 1/60 rather than gaining a share.

B’s old share is \(\frac{2}{3+2}=\frac{2}{5}\), while B’s new share is \(\frac{5}{7+5}=\frac{5}{12}\). Therefore, gaining share = new share − old share = \(\frac{5}{12}-\frac{2}{5}=\frac{25-24}{60}=\frac{1}{60}\). Hence, option A is correct. Exam tip: For a gaining ratio, subtract the old share from the new share; reversing the order gives an incorrect result.

Under the old ratio, A and B each had a share of 4/10. Under the new ratio of 5:5, each receives 5/10. Therefore, A’s gain is 5/10 − 4/10 = 1/10, and B’s gain is also 1/10. Hence, the gaining ratio is 1/10 : 1/10 = 1:1. Remember that 5:5 is the new ratio, not the gaining ratio.

The original total is 6, so A’s old share is 3/6 = 1/2 and B’s old share is 2/6 = 1/3. After C’s death, the new ratio 2:1 gives A a share of 2/3 and B a share of 1/3. A’s gain is therefore 2/3 − 1/2 = 1/6. B’s gain is 1/3 − 1/3 = 0. Thus, only A acquires an additional share, while B acquires none. The gaining ratio is consequently 1/6:0, which is written as 1:0. A zero in the ratio is meaningful here because one continuing partner does not gain any part of C’s share. Therefore, option A is correct.

Gaining ratio is found by subtracting each continuing partner’s old share from the corresponding new share. The old shares are A = 5/10 and B = 3/10. Under the new ratio 3:2, A receives 3/5 = 6/10 and B receives 2/5 = 4/10. A’s gain is 6/10 − 5/10 = 1/10, while B’s gain is 4/10 − 3/10 = 1/10. Therefore, the gaining ratio is 1/10:1/10 = 1:1, making option A correct. Their combined gain is 2/10, exactly equal to C’s former share. Option B is merely the new ratio, while options C and D do not represent the calculated increases.

Under the old ratio, A’s share was 5/10 and B’s share was 3/10. Under the new ratio 3:2, A’s share becomes 6/10 and B’s share becomes 4/10. Thus, A’s gain is 6/10 − 5/10 = 1/10 and B’s gain is 4/10 − 3/10 = 1/10. Therefore, their gaining ratio is 1:1. Exam tip: Gaining ratio = New share − Old share for each continuing partner.

Before C’s retirement, the total ratio is 5, so A and B each hold 2/5. After retirement, the new ratio of A and B is 3:2, giving A 3/5 and B 2/5. A’s share has increased from 2/5 to 3/5, a gain of 1/5. B’s share has remained 2/5, so B’s gain is zero. Consequently, A acquired the greater share of profit; in fact, A alone acquired C’s entire 1/5 share. Option B is incorrect because B’s share did not increase, option C ignores the unequal gains, and option D is false because A clearly gained. Therefore, option A is correct.

Under the old ratio, A’s share was 6/10 and B’s share was 3/10. Under the new ratio, A’s share is 7/10 and B’s share remains 3/10. Thus, A’s gain is 7/10 − 6/10 = 1/10, while B’s gain is 3/10 − 3/10 = 0. Therefore, the gaining ratio is 1/10:0 = 1:0. Exam tip: calculate each partner’s gain by subtracting the old share from the new share.

Under the old ratio 4:1:5, A’s share was 4/10 and B’s share was 1/10. Under the new ratio 3:2, A’s share becomes 3/5 = 6/10 and B’s share becomes 2/5 = 4/10. Thus, both A and B gained a share. Exam tip: In retirement questions, convert the old and new ratios into fractions and compare each continuing partner’s share.

A’s share increases from 4/10 to 3/5, so A gains 6/10 − 4/10 = 2/10. B’s share increases from 1/10 to 2/5, so B gains 4/10 − 1/10 = 3/10. Therefore, the gaining ratio is 2:3.

Under the old ratio, A’s share was 4/10 and B’s share was 1/10. Under the new ratio 3:2, A’s share becomes 6/10 and B’s share becomes 4/10. Therefore, A’s gain is 6/10 − 4/10 = 2/10, while B’s gain is 4/10 − 1/10 = 3/10. Hence, the gaining ratio is 2:3. The ratio 3:2 is the new ratio, not the gaining ratio. Exam tip: Calculate each partner’s gain by subtracting the old share from the new share.