New Ratios

The governing concept is the gaining ratio on retirement. It is calculated by finding how much each continuing partner’s new share exceeds that partner’s old share. The old shares are A = 5/10 = 1/2 and B = 3/10. The new ratio 7:5 totals 12, giving A a new share of 7/12 and B a new share of 5/12. A’s gain is 7/12 − 1/2 = 7/12 − 6/12 = 1/12. B’s gain is 5/12 − 3/10 = 25/60 − 18/60 = 7/60. Therefore, the gaining ratio is 1/12 : 7/60 = 5/60 : 7/60 = 5:7. Option A is correct. Option B is only the new profit-sharing ratio, option C is the old ratio of A and B, and option D incorrectly assumes equal benefit.

C share is 2/10 and A B get 3/20 and 1/20 in 3:1. New shares are 13/20 and 7/20 so the ratio is 13:7.

The sum of the old ratio is 12, so B’s share is \(5/12\). A acquires \(2/5\) of B’s share, which is \(2/12\), while C acquires \(3/5\), which is \(3/12\). Thus, A’s new share is \(4/12+2/12=6/12\), and C’s new share is \(3/12+3/12=6/12\). Therefore, the new ratio is \(6:6=1:1\). The ratio 4:3 ignores the addition of B’s share. Exam tip: Divide the deceased partner’s share according to the acquisition ratio and add the resulting portions to the continuing partners’ old shares.

A’s share is 7/10 of the profits. B and C divide that share in the ratio 4:3, so B gains 4/10 and C gains 3/10. Their new shares are therefore B: 2/10 + 4/10 = 6/10 and C: 1/10 + 3/10 = 4/10. Simplifying 6:4 gives the new ratio 3:2.

A’s old profit share was \(3/(3+4+5)=3/12=1/4\). After C’s retirement, A’s new share is \(5/(5+4)=5/9\). Therefore, A’s gaining share is \(5/9-1/4=(20-9)/36=11/36\). Option B is A’s new share, not the gaining share. Exam tip: Gaining share = New share − Old share.

B’s old profit share was \(\frac{4}{3+4+5}=\frac{4}{12}=\frac{1}{3}\). After C’s retirement, B’s new share is \(\frac{4}{5+4}=\frac{4}{9}\). Therefore, B’s gaining ratio = new share − old share = \(\frac{4}{9}-\frac{1}{3}=\frac{4}{9}-\frac{3}{9}=\frac{1}{9}\). The value \(\frac{4}{9}\) is B’s new share, not the gaining ratio. Exam tip: Calculate a partner’s gaining ratio by subtracting the old share from the new share.

Under the old ratio, A’s share was 3/12 = 1/4 and B’s share was 4/12 = 1/3. Under the new ratio, A’s share is 5/9 and B’s share is 4/9. Therefore, A’s gain = 5/9 − 1/4 = 11/36, while B’s gain = 4/9 − 1/3 = 1/9 = 4/36. Hence, the gaining ratio is 11/36 : 4/36 = 11:4. The ratio 5:4 is the new profit-sharing ratio, not the gaining ratio. Exam tip: To find the gaining ratio, subtract each continuing partner’s old share from the new share.

A old share was 5/10 and new share 2/5 equals 4/10. Therefore A share has decreased and he is not a gainer.

B’s old share was \(4/10\). Under the new ratio 2:3, B’s share is \(3/5=6/10\). Therefore, B’s gaining share = new share − old share = \(6/10-4/10=2/10=1/5\). The value \(1/10\) is C’s old share, not B’s gain. Exam tip: calculate a partner’s gaining share by subtracting the old share from the new share.

Old shares are A = 5/10 = 1/2 and B = 4/10 = 2/5. Their new shares are A = 2/5 and B = 3/5. Thus, A’s share decreases by 1/10, while B’s share increases by 1/5. Therefore, only B gains, so the gaining ratio of A:B is 0:1. The ratio 1:2 compares A’s sacrifice with B’s gain; it is not the gaining ratio because A is not a gaining partner. Exam tip: Subtract each partner’s old share from the new share to identify gain or sacrifice.

The direct answer is A, 17:10. The old total is 4+3+2=9, so A has 4/9, B has 3/9, and C has 2/9. B’s retiring share, 3/9, is acquired by A and C in 5:4. Divide 3/9 into nine equal subparts: A gets 5/9 of 3/9 = 5/27, and C gets 4/9 of 3/9 = 4/27. A’s new share is 4/9 + 5/27 = 12/27+5/27=17/27. C’s new share is 2/9+4/27=6/27+4/27=10/27. Therefore the new ratio is 17:10. Option A is correct. Option B is the old A:B ratio, not the final ratio. Option C is only the acquisition ratio. Option D does not follow from the fractions. Check that 17/27+10/27=1.

A’s share is 6 parts. B and C acquire it in the ratio 1:2, so B receives 2 parts and C receives 4 parts. Therefore, B’s new share is 5+2=7 and C’s new share is 4+4=8, giving a new ratio of 7:8. Exam tip: Add each acquiring partner’s gain to their original share.

The governing concept is that the gaining ratio records the relative additional shares acquired by the continuing partners. The question explicitly states that B and C acquire A’s retiring share in the ratio 3:2. Therefore B receives three units of the additional share for every two units received by C, so their gaining ratio is directly 3:2. No separate subtraction is needed because the acquisition ratio has already been provided. Option B is the old B:C ratio, option C contains A’s old proportion with B and does not measure the gain, and option D would imply equal acquisition, contrary to the stated 3:2 arrangement. Hence option A is unambiguous.

When remaining partners continue in old ratio the gain is also in that ratio. Here A B ratio is 8:3.

After B retires, only A and C will share the profit. Continuing in the old ratio means that the ratio between A and C remains based on their original shares, 9 and 5. Therefore, the new ratio A:C is 9:5. Choosing 6:5 would incorrectly retain the old ratio of B and C. Exam tip: Remove the retiring partner and identify the stated relationship between the remaining partners before forming the new ratio.

After A’s death, only B and C remain as partners. Their respective old profit shares were 8 and 5, so their new ratio remains 8:5. The ratio 3:2 is not a simplified form of 8:5 because 8 and 5 have no common divisor. Exam tip: When the remaining partners agree to continue in their old mutual ratio, that same ratio becomes their new profit-sharing ratio.

A's old profit share was 3/(3+7+5) = 3/15 = 1/5. Under the new ratio of 4:5, A's new profit share is 4/9. Therefore, A's gaining share = new share − old share = 4/9 − 1/5 = (20−9)/45 = 11/45. Option B gives only A's new share, not the gaining share. In such questions, subtract the old share from the new share to find the gaining share.

B’s old share is 7/(3+7+5) = 7/15. After C’s retirement, B’s new share is 5/(4+5) = 5/9. Therefore, B’s gaining share = new share − old share = 5/9 − 7/15 = 25/45 − 21/45 = 4/45. The value 5/9 is B’s new share, not the gaining share. Exam tip: calculate the gaining share by subtracting the old share from the new share.

The old shares of A and B are \(3/15\) and \(7/15\), while their new shares are \(4/9\) and \(5/9\). Therefore, A’s gain is \(4/9-3/15=11/45\), and B’s gain is \(5/9-7/15=4/45\). Hence, the gaining ratio is \(11/45:4/45=11:4\). Remember that the gaining ratio is calculated from the increase in each partner’s share, not by simply stating the new ratio.

A’s old share was 5/14, while the new share is 3/7 = 6/14, so A’s share increased. B’s old share was 5/14, while the new share is 4/7 = 8/14, so B’s share also increased. Therefore, neither A nor B has suffered a decrease; the outgoing partner C’s share has ceased. Exam tip: express the old and new shares with a common denominator before comparing them.

A’s old profit share was \(5/(5+5+4)=5/14\). Under the new ratio 3:4, A’s share is \(3/7=6/14\). Therefore, A’s gaining share = new share − old share = \(6/14-5/14=1/14\). Option B is A’s new share, not the gaining share. Exam tip: Always calculate gaining share as the new share minus the old share.

B’s old profit share was \(5/(5+5+4)=5/14\). After C’s retirement, B’s new share is \(4/(3+4)=4/7=8/14\). Therefore, B’s gaining share = new share − old share = \(8/14-5/14=3/14\). Remember: the gaining share is the increase in a partner’s share, not the partner’s new share itself; hence \(4/7\) is not the answer.

Under the old ratio 5:5:4, A, B and C had shares of 5/14, 5/14 and 4/14 respectively. Under the new ratio 3:4, A and B receive 6/14 and 8/14. Therefore, A gains 6/14 − 5/14 = 1/14, while B gains 8/14 − 5/14 = 3/14. Hence, their gaining ratio is 1:3. In the exam, calculate each partner’s gain as new share minus old share before forming the ratio.

C’s share is 2 out of the total 15 parts. Since A and B acquire it equally, each receives 1 additional part. Thus, A’s new share is 9 + 1 = 10 and B’s is 4 + 1 = 5. Therefore, the new ratio is 10:5 = 2:1. In exams, always express the final ratio in its simplest form.

A’s original share is 2/9. B and C acquire it in the ratio 5:4, so B receives 2/9 × 5/9 = 10/81 and C receives 2/9 × 4/9 = 8/81. B’s new share is 3/9 + 10/81 = 37/81, while C’s new share is 4/9 + 8/81 = 44/81. Therefore, the new ratio is 37:44. Remember that 5:4 is only the acquisition ratio for A’s share, not the final profit-sharing ratio.