New Ratios

Gaining share = New share − Old share. Therefore, A’s gaining share = 1/2 − 1/3 = (3 − 2)/6 = 1/6. Option B represents only the old share, not the gain. Exam tip: always subtract the old share from the new share to calculate the gaining share.

Gaining share = New share − Old share. Therefore, B’s gaining share = 3/8 − 1/4 = 3/8 − 2/8 = 1/8. Option B represents only the old share, not the gain. In such questions, subtract the old share from the new share.

Gaining share = New profit share − Old profit share. Therefore, A’s gaining share = 2/5 − 2/5 = 0. Option B is incorrect because the difference between the two shares is not 1/5. In an exam, remember that a partner has no gain when the old and new shares are equal.

Under the old ratio \(3:2\), A’s share was \(3/5\) and B’s share was \(2/5\). Under the new ratio \(4:1\), their shares become \(4/5\) and \(1/5\), respectively. Thus, A gains \(4/5-3/5=1/5\), while B’s share decreases and B does not gain. Therefore, the gaining ratio of A and B is \(1:0\). Exam tip: a partner whose new share exceeds the old share is a gaining partner.

A’s old share is 1/2 and his new share is 3/5, so his gain is \(3/5-1/2=1/10\). B’s new share is 2/5, which is less than his old share of 1/2; therefore, B is not a gainer. Since only A gains, the gaining ratio is 1:0. Exam tip: the gaining ratio is based on the increase in partners’ shares, not simply on the new ratio.

A's old profit share was \(3/(3+2+1)=3/6=1/2\). Under the new ratio 4:3, A's new share is \(4/(4+3)=4/7\). Therefore, A's gaining share = new share − old share = \(4/7-1/2=(8-7)/14=1/14\). Exam tip: For a retiring partner question, calculate the gaining share by subtracting the old share from the new share.

B’s old profit share was 2/(3+2+1) = 2/6 = 1/3. Under the new ratio 4:3, B’s share is 3/7. Therefore, B’s gaining share = new share − old share = 3/7 − 1/3 = 9/21 − 7/21 = 2/21. Exam tip: Always subtract the old share from the new share to find a partner’s gaining share.

Under the old ratio, A’s share was 3/6 = 1/2 and B’s share was 2/6 = 1/3. Under the new ratio, A’s share is 4/7 and B’s share is 3/7. Therefore, A’s gain = 4/7 − 1/2 = 1/14, while B’s gain = 3/7 − 1/3 = 2/21. Hence, the gaining ratio is 1/14 : 2/21 = 3:4. The ratio 4:3 is the new profit-sharing ratio, not the gaining ratio. Exam tip: Gaining ratio = New share − Old share.

The new ratios of the remaining partners must add up to 1 (100%). After retirement, they share the entire future profit, so their combined share is the whole profit.

The total parts in the new ratio are 3 + 2 = 5. Therefore, the first partner’s share is 3/5 and the second partner’s share is 2/5. Option B is incorrect because writing the ratio directly as 3/2 and 2/3 does not express the shares as fractions of the whole. Exam tip: Add the terms of the ratio and divide each term by their sum.

Since both profit-sharing fractions have the same denominator, their ratio is \(5/8 : 3/8 = 5:3\). Therefore, 5:3 is correct. 3:5 reverses the order of the partners. Exam tip: When fractions have a common denominator, compare their numerators directly to form the ratio.

After A’s death, B and C are the continuing partners. Their former profit-sharing ratio was 3:3, which simplifies to 1:1; therefore, they will share future profits equally. The ratio 4:3 is not the ratio between B and C because it includes A’s original share. Exam tip: Remove the deceased partner’s share and simplify the old ratio of the continuing partners.

The total ratio is 10, so B’s profit share is \(4/10\). This share is acquired equally by A and C, giving each \(2/10\). A’s new share is \(1/10+2/10=3/10\), while C’s new share is \(5/10+2/10=7/10\). Therefore, the new ratio is 3:7. Exam tip: Divide the outgoing partner’s share according to the stated acquisition ratio and add each gain to the partner’s original share.

C’s share is 2/10. A and B acquire it in the ratio 1:3, so A gains 1/20 and B gains 3/20. Their new shares are 2/10 + 1/20 = 5/20 for A and 6/10 + 3/20 = 15/20 for B. Thus, the new ratio is 5:15, or 1:3 in simplest form.

The greatest common divisor of 5 and 15 is 5. Dividing both terms by 5 gives 1:3. Option B is the given ratio, not its simplest form, while option D is also equivalent but not fully simplified. In exams, always divide both terms by their greatest common divisor before writing the final ratio.

The continuing partners compensate the retiring partner for their share of goodwill in the gaining ratio. This ratio shows how much of the retiring partner’s share each continuing partner acquires, so it is used for goodwill adjustment.

After C’s retirement, A and B share profits in the new ratio 7:3. The total of the ratio is 7 + 3 = 10, so A’s new share is 7/10. The option 5/10 may relate to the old ratio, but the question asks for the share according to the new ratio. Exam tip: To find a partner’s share from a ratio, divide that partner’s ratio by the total of all ratio parts.

After retirement, the new ratio is A:B = 7:3. The total parts are 7 + 3 = 10, so B’s new share is 3/10. Option B represents A’s share, 7/10, not B’s share. Exam tip: To find a partner’s share from a new ratio, divide that partner’s ratio part by the sum of all ratio parts.

A’s old profit share was \(5/10\), while the new share is \(7/10\). Therefore, A’s gaining share is \(7/10 - 5/10 = 2/10 = 1/5\). Exam tip: gaining share is calculated as new share minus old share.

B’s old share was \(\frac{3}{10}\), and under the new ratio 7:3, B’s share is still \(\frac{3}{10}\). Therefore, gaining share = new share − old share = \(\frac{3}{10}-\frac{3}{10}=0\). The option \(\frac{3}{10}\) represents B’s new share, not the gain. In such questions, calculate the gaining share by subtracting the old share from the new share.

Under the old ratio, A’s share was 5/10 = 1/2 and B’s share was 3/10. Under the new ratio, A receives 7/10 and B continues to receive 3/10. Thus, A’s gain is 7/10 − 1/2 = 1/5, while B’s gain is 3/10 − 3/10 = 0. Therefore, the gaining ratio is 1/5:0 = 1:0. Exam tip: Calculate each continuing partner’s gain by subtracting the old share from the new share.

The stated new ratio is the ratio in which the remaining partners will share future profits. Compare it with their old shares if you also need to calculate their gaining ratio.

If the remaining partners continue in their old ratio, their relative old shares determine the new ratio. Omit the retiring partner and simplify the remaining shares if possible.

After C retires, C’s share is removed, leaving A and B in the ratio 8:4. Dividing both terms by their greatest common divisor, 4, gives the simplified ratio 2:1. Therefore, option A is correct. Exam tip: When the remaining partners continue in their old ratio, simplify only their original ratio.

The new ratio shows how the remaining partners will share future profits after the retirement or death. A deceased partner’s share may be payable to their legal representative, but that is not the new profit-sharing ratio.