New Ratios

A’s old share was \(6/10=3/5\). B acquires \(3/4\) of A’s share, i.e. \(9/20\), while C acquires \(1/4\), i.e. \(3/20\). Therefore, B’s new share is \(2/10+9/20=13/20\), and C’s new share is \(2/10+3/20=7/20\). Hence, the new ratio is \(13:7\). Exam tip: First calculate the retiring partner’s share, distribute it according to the gaining ratio, and then add the gains to the partners’ old shares.

The relevant concept is redistribution of the retiring partner’s share equally among the continuing partners. The original total is 1+7+2=10, so A’s share is 1/10, B’s is 7/10 and C’s is 2/10. A and C each acquire half of B’s share, namely 7/20. A’s new share becomes 1/10+7/20=2/20+7/20=9/20. C’s new share becomes 2/10+7/20=4/20+7/20=11/20. Hence their new ratio is 9:11, so option A is correct. Option B is only the old A:C ratio; options C and D do not reflect the equal addition of 7/20 to both partners.

The governing rule is that when the remaining partners acquire a deceased partner’s share in their old mutual ratio, their relative proportion does not change. B and C originally have shares 3 and 3, so their old mutual ratio is 3:3, which simplifies to 1:1. A’s 9 parts are transferred between B and C in the same 1:1 relationship; therefore both receive an equal addition. Their individual final shares are equal, even though the total numerical ratio changes from the original three-partner ratio. Thus the new ratio of B and C is 1:1, making option A correct. Option B expresses the same relationship without simplification, while options C and D do not represent the old mutual proportion of B and C.

The given new ratio of A and B is 10:6. Dividing both terms by their greatest common divisor, 2, gives 10 ÷ 2 : 6 ÷ 2 = 5:3. Therefore, the simplified new ratio is 5:3. Although 10:6 represents the same proportion, it is not in its simplest form. In examinations, reduce a ratio to its lowest terms before stating the final answer.

After A’s retirement, B and C share profits in the new ratio 4:3. The total number of parts is 4 + 3 = 7, so B’s new share is 4/7. Option B, 3/7, represents C’s share, while 4/3 is the ratio itself, not B’s share. Exam tip: To find a partner’s share, divide that partner’s ratio term by the sum of all ratio terms.

Before C’s retirement, A’s share was 4/10 and B’s share was 5/10. Under the new 1:1 ratio, each partner receives 1/2, or 5/10. Therefore, A gains 5/10 − 4/10 = 1/10, while B’s share remains unchanged at 5/10. Hence, only A gained a share. Exam tip: Convert both the old and new ratios into fractions of the total profit before comparing them.

A moved from 3/10 to 2/5 or 4/10 and B moved from 5/10 to 3/5 or 6/10. Both gained 1/10 equally.

The governing concept is gaining ratio, which is calculated by comparing each continuing partner’s new share with the old share. The original total is 10, so A’s old share is 2/10 and B’s old share is 5/10. Their new ratio 3:7 has a total of 10, giving A a new share of 3/10 and B a new share of 7/10. A’s gain is 3/10−2/10=1/10, and B’s gain is 7/10−5/10=2/10. Therefore their gains are 1/10:2/10, which simplifies to 1:2. Option B is the new profit-sharing ratio, not the gain in shares; options C and D are respectively the old ratio and an unsupported equal division.

The governing concept is the gaining ratio, which measures how much each continuing partner’s share increases after a partner retires. The old ratio has 10 parts, so A’s old share is 4/10 and B’s old share is 1/10. The new ratio gives A 7/10 and B 3/10. Therefore, A’s gain is 7/10 − 4/10 = 3/10, while B’s gain is 3/10 − 1/10 = 2/10. Removing the common denominator gives the gaining ratio A:B = 3:2, so option A is correct. The gains total 5/10, exactly equal to C’s former share. Option B is merely the new ratio, option C is the old ratio, and option D wrongly assumes equal gains.

Under the old ratio 6:2:2, A’s share was 6/10 = 3/5 and B’s share was 2/10 = 1/5. Under the new ratio 4:1, A receives 4/5 and B receives 1/5. Therefore, A’s gain is 4/5 − 3/5 = 1/5, while B’s gain is 1/5 − 1/5 = 0. Hence, the gaining ratio of A and B is 1:0. Exam tip: Calculate each partner’s gain by subtracting the old share from the new share.

A and C’s old mutual ratio is 1:4. B’s share is 5/10, and they acquire it in the ratio 1:4. Therefore, A gains 1/10 and C gains 4/10. Their new shares become 1/10 + 1/10 = 2/10 and 4/10 + 4/10 = 8/10, giving 2:8 = 1:4. Exam tip: When the retiring partner’s share is acquired by the remaining partners in their old mutual ratio, their new ratio remains the same mutual ratio.

B share is 3/10. A gets 2/10 and C gets 1/10 so new shares are 6/10 and 4/10 with simplified ratio 3:2.

A share is 2/10 and equal taking gives B and C 1/10 each. New shares are 8/10 and 2/10 so the ratio is 4:1.

After retirement only continuing partners run the firm. Therefore a new ratio is required for future profits and losses.

A’s old share was 3/(3+1+6) = 3/10. After C’s retirement, A’s new share is 2/(2+1) = 2/3. Therefore, A’s gain = new share − old share = 2/3 − 3/10 = 20/30 − 9/30 = 11/30. Thus, 2/3 is A’s new share, not the gain. Exam tip: calculate a partner’s gain as new share minus old share.

B’s old profit share was \(1/10\), since B had 1 part out of a total of 10 parts in the old ratio 3:1:6. In the new ratio 2:1, B’s share is \(1/3\). Therefore, B’s gain ratio = new share − old share = \(1/3 - 1/10 = 7/30\). Exam tip: calculate a partner’s gain by subtracting the old share from the new share.

Under the old ratio, A's share was 3/10 and B's share was 1/10. Under the new 2:1 ratio, A's share becomes 2/3 and B's share becomes 1/3. Thus, A's gain = 2/3 − 3/10 = 11/30, and B's gain = 1/3 − 1/10 = 7/30. Therefore, the gaining ratio is 11/30 : 7/30 = 11:7. Option 2:1 is the new ratio, not the gaining ratio. Exam tip: calculate each partner's gain by subtracting the old share from the new share.

The correct answer is Option A, 2:3:2. The old ratio A:B:C:D is 2:3:2:3. D retires, so D's share is removed. No information is given about any change in the relative claims of A, B and C. Therefore the remaining partners continue in their old mutual ratio, namely A:B:C = 2:3:2. Option A is correct. Option B, 2:3:3, increases C's share without any stated reason. Option C, 1:1:1, makes all three remaining partners equal, but the question gives no such agreement. Option D, 3:2:3, changes the positions of A and B and is not supported. It is not necessary to divide the remaining total 7 parts into a new numerical base; a ratio may be simplified only if all terms have a common factor, and 2:3:2 cannot be simplified. The safest rule is to remove the retiring partner's term and preserve the other terms in order. Memory cue: no information means no change in the continuing partners' mutual ratio.

Since B is the retiring partner, B’s share of 2 is omitted. The original shares of the continuing partners A, C and D are 5, 4 and 1 respectively, so their old mutual ratio is 5:4:1. Option B is incorrect because it still includes the retiring partner B. Exam tip: For the old mutual ratio, retain only the original proportional shares of the continuing partners; there is no need to convert them into a new total of 1.

The total ratio is 2+2+3+3=10, so D’s old share was 3/10. Since D’s share is acquired equally by A, B and C, each receives 3/10 ÷ 3 = 1/10. Therefore, A’s new share is 2/10 + 1/10 = 3/10. Option B is only A’s old share and does not include the share acquired from D. Exam tip: Always add the retiring partner’s acquired share to the continuing partner’s old share.

The old ratio is 2:2:3:3, so D’s share is 3/10. This share is acquired equally by A, B and C; therefore, each receives 1/10. A’s new share is 2/10 + 1/10 = 3/10, B’s is 3/10, and C’s is 3/10 + 1/10 = 4/10. Hence, the new ratio is 3:3:4. Option B is merely the old ratio and does not account for the equal distribution of D’s share. Exam tip: Divide the retiring partner’s share equally among the gaining partners and add each gain to their old share.

The governing principle is that the retiring partner’s old share is distributed among the continuing partners in the stated acquisition ratio, and each gain is added to the corresponding old share. The old total is 3+2+1+4=10, so C’s share is 1/10. The acquisition ratio 2:1:1 has four parts. A receives 2/4 of 1/10 = 1/20, while B and D each receive 1/40. Thus A’s new share is 3/10+1/20=14/40; B’s is 2/10+1/40=9/40; and D’s is 4/10+1/40=17/40. Hence the new ratio is 14:9:17, making option A correct. The sum is 40/40, confirming the result.

B share is 3/10 and when divided in 2:1:3 the new shares become 4/20 5/20 and 11/20. Therefore the new ratio is 4:5:11.

A share moved from 5/10 to 3/5 or 6/10 so gain is 1/10. B share remained 4/10 so his gain is zero.

Assume the total profit is divided into 10 parts. A’s original share is 3/10 and C’s is 2/10. B’s share of 5/10 is acquired by A and C in the ratio 4:1, so A receives 4/10 and C receives 1/10. Therefore, A’s new share is 3/10 + 4/10 = 7/10, while C’s new share is 2/10 + 1/10 = 3/10. Hence, the new ratio is 7:3. Exam tip: Divide the retiring partner’s share according to the acquisition ratio and add the resulting portions to the partners’ original shares. The ratio 4:1 is only the acquisition ratio, not the final profit-sharing ratio.