New Ratios

Correct answer: A. (5:7)

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.

Correct answer: A. 13:7

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.

Correct answer: A. 1:1

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.

Correct answer: A. 3:2

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.

Correct answer: A. 11/36

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.

Correct answer: A. 1/9

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.

Correct answer: A. 11:4

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.

Correct answer: A. A

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.

Correct answer: A. 1/5

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.

Correct answer: A. 0:1

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.

Correct answer: A. 17:10

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.

Correct answer: A. 7:8

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.

Correct answer: A. 3:2

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.

Correct answer: A. 8:3

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

Correct answer: A. 9:5

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.

Correct answer: A. 8:5

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.

Correct answer: A. 11/45

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.

Correct answer: A. 4/45

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.

Correct answer: A. 11:4

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.

Correct answer: D. Neither A nor B

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.

Correct answer: A. 1/14

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.

Correct answer: A. 3/14

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.

Correct answer: A. 1:3

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.

Correct answer: A. 2:1

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.

Correct answer: A. 37:44

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.