New Ratios

B’s share consists of 7 parts, which A and C acquire in the ratio 3:4. Thus, A receives 3 parts and C receives 4 parts from B’s share. A’s new share is 6+3=9, while C’s new share is 2+4=6. Therefore, the new ratio is 9:6, which simplifies to 3:2. The option 9:4 is incorrect because C’s original share of 2 parts must also be included. Exam tip: Divide the outgoing partner’s share in the stated acquisition ratio and add each acquired portion to the respective partner’s original share.

Gaining ratio = New share − Old share. The first partner’s gain is \(7/12-5/12=2/12\). The second partner’s old share is \(1/4=3/12\), so the gain is \(5/12-3/12=2/12\). Therefore, the gaining ratio is \(2/12:2/12=2:2\), which is also expressible in simplified form as \(1:1\). In an exam, calculate each partner’s gain separately; using the ratio of new shares alone would be incorrect.

Gaining share = New share − Old share. Convert the old share to a common denominator: \(\frac{4}{9}=\frac{8}{18}\). Therefore, A's gaining share is \(\frac{11}{18}-\frac{8}{18}=\frac{3}{18}=\frac{1}{6}\). The value \(\frac{11}{18}\) is the new share itself, not the gain. Exam tip: Always subtract the old share from the new share to find a partner's gaining share.

Gaining ratio = New ratio − Old ratio = \(9/20 - 7/20 = 2/20 = 1/10\). Therefore, B’s gaining ratio is \(1/10\). The value \(9/20\) represents B’s new ratio, not the gain. Exam tip: calculate a partner’s gain by subtracting the old ratio from the new ratio.

The new ratio 8:7 has a total of 15, so the new shares are 8/15 and 7/15. The first partner’s gain is 8/15 − 1/2 = 1/30, while the second partner’s gain is 7/15 − 1/3 = 4/30. Therefore, the gaining ratio is 1/30 : 4/30 = 1:4. Option B reverses the ratio, while 8:7 is the new ratio, not the gaining ratio. Exam tip: Calculate new share − old share for each partner and then simplify the gains.

The direct answer is A, 2:0. The old total is 5+5+2=12. Thus A and B each originally had 5/12. After C retires, A:B is 7:5, whose total is 12, so A’s new share is 7/12 and B’s new share is 5/12. A’s gain is 7/12−5/12=2/12; B’s gain is 5/12−5/12=0. Therefore the gaining ratio is 2:0, equivalently 1:0. Option A is correct. Option B is the new profit-sharing ratio, not the gaining ratio. Option C is the old A:B ratio, not the gains. Option D incorrectly says both gained equally; only A received an increase. A zero gain is written in the ratio because B’s share did not change. Remember: gaining ratio uses increases, not simply the final ratio.

Before C's retirement, A's share was 7/10 and B's share was 1/10. Under the new ratio of 3:2, A's share becomes 3/5, or 6/10, while B's share becomes 2/5, or 4/10. Thus, A's share decreases from 7/10 to 6/10, whereas B's share increases. Exam tip: Convert the old and new ratios into fractions with a common denominator before comparing them.

B’s old profit share was \(1/10\). Under the new ratio of 3:2, B’s new share is \(2/5=4/10\). Therefore, B’s gaining ratio = new share − old share = \(4/10-1/10=3/10\). Option B, \(2/5\), represents B’s new share, not the gain. In such questions, calculate gaining ratio as ‘new share − old share’.

Under the old ratio, A’s share was \(7/10\) and B’s share was \(1/10\). Under the new ratio 3:2, A’s share is \(3/5=6/10\) and B’s share is \(2/5=4/10\). Thus, A’s share decreases by \(1/10\), while B’s share increases by \(3/10\). Therefore, A gains nothing and B gains \(3/10\), giving a gaining ratio of \(0:3=0:1\). Exam tip: The gaining ratio includes only the increases in the shares of the continuing partners who actually gain.

A’s old profit share was 4/(4+7+4) = 4/15. Under the new ratio 3:2, A’s new share is 3/5 = 9/15. Therefore, A’s gaining share = new share − old share = 9/15 − 4/15 = 5/15 = 1/3. Option B is A’s new share, not the gain. Exam tip: Always calculate gaining share as the new share minus the old share.

C’s old profit share was \(4/(4+7+4)=4/15\). Under the new ratio 3:2, C’s new share is \(2/(3+2)=2/5=6/15\). Therefore, C’s gaining share = new share − old share = \(6/15-4/15=2/15\). Exam tip: do not confuse the new share \(2/5\) with the gaining share; gaining share is always calculated as new share minus old share.

Under the old ratio, A and C each had a share of 4/15. Under the new ratio of 3:2, A’s share is 3/5 = 9/15 and C’s share is 2/5 = 6/15. Therefore, A’s gain is 9/15 − 4/15 = 5/15, while C’s gain is 6/15 − 4/15 = 2/15. Hence, the gaining ratio is 5:2. Exam tip: Gaining share = new share − old share.

The correct answer is Option A, Extra shares were acquired in 2:1. The new ratio 4:3 tells how the continuing partners will divide the entire future profit after reconstitution. The gaining ratio 2:1 tells only the proportion in which the partners acquired the retiring partner's surrendered share. These two ratios answer different questions and need not be equal. Option A is correct because a gaining ratio of 2:1 directly means that the extra shares were obtained in the proportion 2:1. Option B is wrong because the total future profit is divided in the new ratio 4:3, not in the gaining ratio 2:1. Option C is wrong because new ratio and gaining ratio are not always the same; they may happen to look similar in some examples, but their meanings differ. Option D is wrong because a retiring partner does give up a share, and the gaining partners acquire it. A common mistake is to treat every ratio in the question as a profit-sharing ratio. Exam cue: “new” means total future shares; “gaining” means only the additional shares.

In the new ratio 11:9, A’s new share is 11/(11+9) = 11/20. A’s old share, 1/2, is equal to 10/20. Therefore, A’s gaining share = new share − old share = 11/20 − 10/20 = 1/20. Remember that the gaining share is found by subtracting the old share from the new share; 11/20 is only the new share.

B’s new share is \(\frac{9}{11+9}=\frac{9}{20}\). B’s old share, \(\frac{2}{5}\), is equal to \(\frac{8}{20}\). Therefore, B’s gaining share = new share − old share = \(\frac{9}{20}-\frac{8}{20}=\frac{1}{20}\). Option C is only B’s old share, not the gain. Exam tip: For a gaining partner, subtract the old share from the new share.

After D’s retirement, the old mutual ratio of A, B and C remains 4:3:2. Therefore, their new ratio is 4:3:2. D’s 1/10 share is not redistributed among them because the question states that they continue in their old mutual ratio. Exam tip: When continuing partners share profits in their old mutual ratio, simply omit the retiring partner’s term from the original ratio.

C’s old share is 3 out of a total of 14 parts, i.e. 3/14. Since this share is acquired equally by A, B and D, each receives 1/14. Therefore, A’s new share is 5/14+1/14=6/14, B’s is 4/14+1/14=5/14, and D’s is 2/14+1/14=3/14. Hence, the new ratio is 6:5:3. Exam tip: Express the retiring partner’s share in terms of the total old parts before distributing it according to the acquisition ratio.

The total of the old ratio is 18, so B’s share is 5/18. This share is acquired by A, C and D in the ratio 2:2:1; therefore, A receives 2/18, C receives 2/18 and D receives 1/18. Their new shares are A = 6/18 + 2/18 = 8/18, C = 4/18 + 2/18 = 6/18 and D = 3/18 + 1/18 = 4/18. Thus, the new ratio is 8:6:4, which simplifies to 4:3:2. In an examination, always express the final ratio in its simplest form.

A’s original share was 3 out of the total 14 parts, or 3/14. This share is acquired by B, C and D in the ratio 2:1:0. Thus, B gains 2/14, C gains 1/14 and D gains nothing. Their new shares are B = 4/14 + 2/14 = 6/14, C = 5/14 + 1/14 = 6/14 and D = 2/14. Therefore, the new ratio is 6:6:2. Exam tip: Add the retiring partner’s share to each continuing partner’s old share according to the gaining ratio.

D’s share is 2/14 of the total profit. This share is acquired by A, B and C in the ratio 3:2:1, so their respective gains are 1/14, 1/21 and 1/42. Their new shares are A = 7/14 + 1/14 = 24/42, B = 3/14 + 1/21 = 11/42, and C = 2/14 + 1/42 = 7/42. Therefore, the simplified new ratio is 24:11:7. Exam tip: First calculate the retiring partner’s share, distribute it in the gaining ratio, and add each gain to the partner’s old share.

The governing concept is the calculation of a new profit-sharing ratio after retirement. The total of the old ratio is 16, so A’s share is 8/16 = 1/2. This retiring share is acquired equally by three continuing partners; therefore each receives 1/2 × 1/3 = 1/6. B’s new share is 4/16 + 1/6 = 5/12, C’s is 3/16 + 1/6 = 25/48, and D’s is 1/16 + 1/6 = 11/48. Expressing all shares over 48 gives B = 20/48, C = 16/48 and D = 11/48. Hence the new ratio is 20:16:11, so option A is correct. The other options either ignore equal acquisition or use incorrect additions.

D share of 3 parts is divided in 1:2:3. A gets 0.5 B gets 1 and C gets 1.5 so the ratio is 5:12:11.

After retirement or death total future profit must equal one whole share. If total is more than one the calculation is wrong.

The sum of the terms in the new ratio 47:28 is 47 + 28 = 75. Therefore, the first partner’s new share is 47/75 and the second partner’s new share is 28/75. Option C reverses the partners’ shares, while option B does not convert the ratio into proper fractions. Exam tip: To convert a ratio into shares, divide each term by the sum of all the terms.

The new profit-sharing ratio is obtained by comparing the partners’ shares: 29/50 : 16/50. Since both fractions have the same denominator, the denominator cancels, giving the ratio 29:16. Therefore, option A is correct. Exam tip: when the denominators are equal, compare the numerators directly.