New Ratios

B’s original share is 3 parts. A and C acquire this share in the ratio 2:1, so A receives 2 parts and C receives 1 part. Therefore, A’s new share is 6+2=8 and C’s new share is 1+1=2. Hence, the new ratio of A:C is 8:2, which simplifies to 4:1. Exam tip: Add the deceased partner’s share to the gaining partners in their gain ratio, then simplify the resulting ratio.

Gain is calculated as new share minus old share. The first partner’s gain is 5/8 − 3/8 = 2/8, and the second partner’s gain is 3/8 − 1/4 = 3/8 − 2/8 = 1/8. Therefore, the gaining ratio is 2/8 : 1/8 = 2:1. Exam tip: convert fractions to a common denominator before subtracting, and check that each new share exceeds the corresponding old share.

Gaining share = New share − Old share. Since \(\frac{1}{3}=\frac{4}{12}\), A’s gaining share is \(\frac{7}{12}-\frac{4}{12}=\frac{3}{12}=\frac{1}{4}\). \(\frac{7}{12}\) is only the new share, not the gain. Exam tip: Convert fractions to a common denominator before subtracting.

Gaining ratio = New ratio − Old ratio. Therefore, B’s gaining ratio is \(7/16 - 5/16 = 2/16 = 1/8\). Options C and D are the old and new ratios respectively, not the gain. Exam tip: Always subtract the old ratio from the new ratio to find a partner’s gain.

The total of the new ratio 7:5 is 12, so the new shares are 7/12 and 5/12. The first partner’s gain is 7/12 − 1/2 = 1/12, and the second partner’s gain is 5/12 − 1/3 = 1/12. Therefore, the gaining ratio is 1/12:1/12 = 1:1. Option B is the new ratio, not the gaining ratio. Exam tip: Calculate each partner’s gain by subtracting the old share from the new share.

Before C’s retirement, A and B each had a share of \(4/10\). Under the new ratio of 5:5, each receives \(1/2=5/10\). Thus, A’s gain is \(5/10-4/10=1/10\), and B’s gain is also \(1/10\). Therefore, their gaining ratio is \(1/10:1/10=1:1\). In the exam, calculate each partner’s gain by subtracting the old share from the new share.

The total of the old ratio is 10, so A’s old share was 6/10. In the new ratio 5:5, A’s share is 5/10, or 1/2. Since 5/10 is less than 6/10, A’s profit share decreased. B’s share, in contrast, increased from 2/10 to 5/10. Exam tip: Convert the old and new shares to a common denominator before comparing them.

B’s old profit share was 2/10. Under the new ratio 5:5, B’s new share is 5/10, or 1/2. Therefore, B’s gaining share = new share − old share = 5/10 − 2/10 = 3/10. Option 1/2 represents B’s new share, not the share gained. Exam tip: calculate gaining share as new share minus old share.

Under the old ratio, A’s share was 6/10 and B’s share was 2/10. Under the new ratio of 5:5, each receives 5/10. Thus, A’s share decreases by 1/10, while B’s share increases by 3/10. Therefore, the gaining ratio of A:B is 0:3, which simplifies to 0:1. Exam tip: Calculate gain as new share minus old share, and include only the partners whose shares increase.

The total of the old ratio 3:5:2 is 10, so A’s old share was 3/10. After B’s retirement, A’s new share is 7/10. Therefore, A’s gaining share = new share − old share = 7/10 − 3/10 = 4/10 = 2/5. The value 7/10 is A’s new share, not the gaining share. Exam tip: Always calculate the gaining share by subtracting the old share from the new share.

The old total ratio is \(3+5+2=10\), so C’s old share was \(\frac{2}{10}\). Under the new ratio 7:3, C’s new share is \(\frac{3}{10}\). Therefore, C’s gaining share = new share − old share = \(\frac{3}{10}-\frac{2}{10}=\frac{1}{10}\). The value \(\frac{3}{10}\) is C’s new share, not the gaining share. Exam tip: calculate a gaining share by subtracting the old share from the new share.

Under the old ratio, A’s share was 3/10 and C’s share was 2/10. Under the new ratio, A’s share is 7/10 and C’s share is 3/10. Therefore, A’s gain is 7/10 − 3/10 = 4/10, while C’s gain is 3/10 − 2/10 = 1/10. Hence, the gaining ratio is 4/10 : 1/10 = 4:1. Exam tip: calculate each continuing partner’s gain by subtracting the old share from the new share.

The gaining ratio is the ratio in which the continuing partners acquire the additional share surrendered by the retiring partner. A ratio of 1:1 means that both partners acquired equal additional shares. The new ratio, 3:2, represents their total future profit shares, which are not equal. Therefore, option A is correct. Exam tip: distinguish total new shares from the additional shares gained from the retiring partner.

Under the new ratio 9:7, A's new share is 9/(9+7) = 9/16. A's old share, 1/2, is equal to 8/16. Therefore, A's gaining share = new share − old share = 9/16 − 8/16 = 1/16. The value 9/16 is A's new total share, not the gain. Exam tip: Always calculate gaining share by subtracting the old share from the new share.

In the new ratio 9:7, the total number of parts is 16, so B’s new share is \(7/16\). B’s old share \(3/8\) is equal to \(6/16\). Therefore, B’s gaining share = new share − old share = \(7/16-6/16=1/16\). Option C is only B’s old share, not the gain. Exam tip: Calculate gaining share as new share minus old share.

D’s retirement removes only D’s share from the partnership. Since A, B and C agree to continue sharing profits in their old ratio, their new ratio remains 3:2:2. In option B, C’s share has been incorrectly reduced to 1. Exam tip: when the continuing partners retain their old ratio, write the old ratio of only those continuing partners.

The correct answer is Option A, 5:3.5:1.5. The old ratio A:B:C:D is 4:3:2:1. C retires and gives up 2 parts. A, B and D acquire those 2 parts in the ratio 2:1:1. The total units in the acquiring ratio are 4, so one unit equals 2/4 = 0.5 of C's surrendered share. A receives 2 units = 1 part; B receives 1 unit = 0.5 part; D receives 1 unit = 0.5 part. Add these gains to their old shares: A = 4 + 1 = 5, B = 3 + 0.5 = 3.5, and D = 1 + 0.5 = 1.5. Hence the new ratio is 5:3.5:1.5. Option A is correct. Option B, 6:4:2, gives a different total distribution and does not follow the stated acquisition ratio. Option C, 4:3:1, gives D no acquired share and is therefore wrong. Option D, 2:1:1, is only the acquisition ratio, not the total new profit-sharing ratio. Decimals may be removed by multiplying all terms by 2, giving 10:7:3, but the listed form 5:3.5:1.5 is valid.

C’s share is 2 out of 10 parts, or 1/5. A, B and D acquire this share in the ratio 2:1:1, so their gains are 1/10, 1/20 and 1/20 respectively. Their new shares are A = 4/10 + 1/10 = 1/2, B = 3/10 + 1/20 = 7/20, and D = 1/10 + 1/20 = 3/20. Multiplying these by 20 gives the simplified ratio 10:7:3. Exam tip: First express the retiring partner’s share as a fraction, then distribute it according to the gaining ratio.

The old total ratio is 5+4+3+2=14, so B’s share is 4/14. This share is acquired by A, C and D in the ratio 1:2:1; therefore, they receive additional shares of 1/14, 2/14 and 1/14, respectively. The new shares are A = 5/14+1/14=6/14, C = 3/14+2/14=5/14 and D = 2/14+1/14=3/14. Hence, the new ratio is 6:5:3. The ratio 1:2:1 represents only the acquisition of B’s share, not the new total profit-sharing ratio. Exam tip: Add each partner’s acquired portion to their existing share.

A’s share is 6 out of the total 15 parts. This share is divided equally among B, C and D, so each receives 2 parts. Therefore, the new ratio is B:C:D = (4+2):(3+2):(2+2) = 6:5:4. Exam tip: add the acquired share to each gaining partner’s old share.

The total of the old ratio is 2+3+4+5=14, so D’s share is 5/14. This share is acquired by A, B and C in the ratio 2:2:1, giving A 2/14, B 2/14 and C 1/14. Therefore, the new shares are A = 2/14+2/14=4/14, B = 3/14+2/14=5/14, and C = 4/14+1/14=5/14. Hence, the new ratio is 4:5:5. Exam tip: Express the retiring partner’s share in terms of the total old ratio before distributing it in the gaining ratio.

After retirement or death the whole future profit must be shared by remaining partners. So total new share should be 1.

The total of the new ratio 13:7 is 13 + 7 = 20. Therefore, the first partner’s share in the total profit is 13/20 and the second partner’s share is 7/20. Option C reverses the partners’ shares, while option B gives reciprocal values rather than shares. Exam tip: To convert a ratio into shares, divide each ratio term by the sum of all ratio terms.

The new ratio is obtained by comparing the partners’ new profit-sharing fractions. Thus, \(\frac{17}{27}:\frac{10}{27}=17:10\), because the common denominator 27 cancels out. Therefore, option A is correct. Exam tip: When fractions have the same denominator, their ratio is simply the ratio of their numerators.

New ratio is the ratio of entire future profit of remaining partners. Gaining ratio shows only acquisition of retiring share.