New Ratios

Under the old ratio, A’s share was 9/15 and B’s share was 5/15. Under the new ratio of 2:1, A’s share becomes 10/15 and B’s share remains 5/15. Therefore, A’s gain is 10/15 − 9/15 = 1/15, while B’s gain is 5/15 − 5/15 = 0. Hence, the gaining ratio is 1/15:0 = 1:0. Exam tip: Gaining ratio is calculated as new share minus old share; do not subtract the retiring partner’s share separately from each continuing partner.

D’s original share is 2/10. When this share is acquired by A, B and C in the ratio 2:3:5, their respective gains are 4/100, 6/100 and 10/100. Therefore, their new shares are 30/100+4/100=34/100, 30/100+6/100=36/100 and 20/100+10/100=30/100. Thus, the new ratio is 34:36:30, which simplifies to 17:18:15. Exam tip: distribute the retiring partner’s share according to the stated acquisition ratio, then add each gain to the partner’s old share.

A’s share is 6 out of 10, or 6/10. When this share is acquired in the ratio 1:3:2, B, C and D receive 1/10, 3/10 and 2/10 respectively. Their new shares are therefore B = 1/10 + 1/10 = 2/10, C = 2/10 + 3/10 = 5/10, and D = 1/10 + 2/10 = 3/10. Hence, the new ratio is 2:5:3. The ratio 1:3:2 represents only the manner in which A’s share is acquired; it is not the final ratio. Exam tip: add each partner’s gain from the deceased partner’s share to their original share.

A’s old share was 2/(2+4+4) = 2/10. After C’s retirement, A’s new share is 3/(3+7) = 3/10. Therefore, A’s gain = new share − old share = 3/10 − 2/10 = 1/10. Option B represents A’s new share, not the gain. Exam tip: Always calculate gain by subtracting the old share from the new share.

B’s old share was 4/(2+4+4) = 4/10. After C’s retirement, B’s new share is 7/(3+7) = 7/10. Therefore, B’s gain = new share − old share = 7/10 − 4/10 = 3/10. The value 4/10 is B’s old share, not the gain. Exam tip: Always calculate a partner’s gain as new share minus old share.

Under the old ratio, A’s share was \(2/10\) and B’s share was \(4/10\). Under the new ratio of 3:7, their shares are \(3/10\) and \(7/10\), respectively. Thus, A gains \(1/10\) and B gains \(3/10\), giving a gaining ratio of \(1:3\). Exam tip: Calculate gain as new share minus old share, then reduce the gains to their simplest ratio.

Under the old ratio 3:6:1, A's share was 3/10 and B's share was 6/10. Under the new ratio 4:6, A's share becomes 4/10 while B's share remains 6/10. Thus, A gains 1/10 of the profit, whereas B's share is unchanged. Therefore, only A gains a share from retiring partner C. Exam tip: Convert the old and new ratios to fractions with the same denominator before comparing individual shares.

Under the old ratio, A’s share was 3/10 and B’s share was 6/10. Under the new ratio of 4:6, A’s share is 4/10 and B’s share remains 6/10. Thus, A gains 1/10 while B gains nothing, giving a gaining ratio of 1:0. The ratio 4:6 is the new profit-sharing ratio, not the gaining ratio. Exam tip: Gaining ratio = New ratio − Old ratio.

The direct answer is option A, 3:2. The old ratio A:B:C is 2:1:7, so the total is 10 parts. C retires and gives up 7/10 of the profit. A and B acquire that 7/10 in the ratio 4:3. The seven parts are divided into 4 + 3 = 7, so A gets 4/10 and B gets 3/10. A's new share is 2/10 + 4/10 = 6/10. B's new share is 1/10 + 3/10 = 4/10. Thus the new ratio is 6:4, which reduces by 2 to 3:2. Option A is correct. Option B, 4:3, is only the gaining ratio, not the final profit-sharing ratio. Option C, 2:1, is the old ratio of A and B and ignores C's transferred share. Option D, 6:4, represents the unreduced ratio; it is numerically equivalent to 3:2, but in a single-best-answer question the simplified form in option A is the expected answer. Exam cue: distinguish the gaining ratio from the new ratio, then simplify the final parts.

The total ratio is 10, so A’s old share is 2/10 and B’s share is 5/10. A acquires \(\frac{2}{5}\) of B’s share, i.e. \(\frac{2}{5}\times\frac{5}{10}=\frac{2}{10}\). Therefore, A’s new share is \(\frac{2}{10}+\frac{2}{10}=\frac{4}{10}=\frac{2}{5}\). Exam tip: after a partner’s retirement, add the share acquired from the retiring partner to the continuing partner’s old share.

The total of the original ratio is 10, so B’s share is 5/10. This share is distributed among A, C and D in the ratio 2:2:1, giving them additional shares of 2/10, 2/10 and 1/10, respectively. Therefore, A’s new share is 2/10 + 2/10 = 4/10, C’s is 1/10 + 2/10 = 3/10, and D’s is 2/10 + 1/10 = 3/10. Hence, the new ratio is 4:3:3. Exam tip: Divide the retiring partner’s share according to the gaining ratio and add each gain to the partner’s original share.

A’s old profit share was 1/(1+4+5) = 1/10. After C’s retirement, A’s new share is 2/(2+3) = 2/5 = 4/10. Therefore, A’s gain = new share − old share = 4/10 − 1/10 = 3/10. The option 2/5 represents A’s new share, not the gain. Exam tip: Always calculate a partner’s gain as new share minus old share.

B’s old share was 4/10. Under the new ratio of 2:3, B’s new share is 3/5 = 6/10. Therefore, B’s gain = new share − old share = 6/10 − 4/10 = 2/10 = 1/5. Option C represents B’s old share, not the gain. Exam tip: Always calculate a partner’s gain by subtracting the old share from the new share.

Under the old ratio, A’s share was 1/10 and B’s share was 4/10. Under the new ratio of 2:3, A’s share becomes 4/10 and B’s share becomes 6/10. Therefore, A’s gain is 4/10 − 1/10 = 3/10, while B’s gain is 6/10 − 4/10 = 2/10. Hence, the gaining ratio is 3:2. Remember that 2:3 is the new ratio, not the gaining ratio; the gaining ratio is calculated from the increase in each continuing partner’s share.

A’s original share is 3/10. B, C and D acquire it in the ratio 2:1:3, so their gains are 1/10, 1/20 and 3/20 respectively. Therefore, their new shares are B = 2/10 + 1/10 = 3/10, C = 3/10 + 1/20 = 7/20, and D = 2/10 + 3/20 = 7/20. Thus, the new ratio is 3/10:7/20:7/20 = 6:7:7. Exam tip: Divide the deceased partner’s share in the gaining ratio and add each gain to the partner’s original share.

The main task in new ratio is to understand distribution of retiring partner share. First identify retiring share and acquisition ratio.

If no information is given the old mutual ratio of continuing partners becomes the new ratio. So A and B ratio remains 7:4.

After removing B the old shares of A C and D are 3:2:4. If no information is given this will be the new ratio.

C share is 1/10 and equal acquisition gives 1/20 each to A and B. New shares are 13/20 and 7/20.

The total ratio is 4+5+3=12, so C’s share is 3/12. A acquires one-third of C’s share, namely 1/12, while B acquires two-thirds, namely 2/12. Therefore, A’s new share is 4/12+1/12=5/12 and B’s new share is 5/12+2/12=7/12. Hence, the new ratio is 5:7. Option 4:5 is the old ratio, and 6:8 is not equivalent to 5:7. Exam tip: Divide the retiring partner’s share according to the acquisition ratio and add the respective portions to the old shares.

B’s share of the total profit is \(7/10\). Since A and C acquire it in the ratio 3:4, A receives \(3/7 \times 7/10=3/10\) and C receives \(4/7 \times 7/10=4/10\). Thus, A’s new share is \(2/10+3/10=5/10\), while C’s new share is \(1/10+4/10=5/10\). Therefore, the new ratio is 1:1. The ratio 3:4 is only the ratio in which B’s share is acquired, not the final profit-sharing ratio. Exam tip: first calculate the retiring partner’s share, then distribute it among the gaining partners.

A’s original share is \(5/10\). B and C acquire it in the ratio 2:3, so B receives \(2/5 \times 5/10 = 2/10\) and C receives \(3/5 \times 5/10 = 3/10\). Therefore, B’s new share is \(2/10+2/10=4/10\), while C’s new share is \(3/10+3/10=6/10\). The new ratio is 4:6, which simplifies to 2:3. Exam tip: Apply the acquisition ratio to the deceased partner’s original share, then add the gain to each continuing partner’s old share.

A’s old share = 3/(3+1+6) = 3/10. After C’s retirement, A and B share profits in the new ratio 2:1, so A’s new share = 2/(2+1) = 2/3. Therefore, A’s gain = new share − old share = 2/3 − 3/10 = 20/30 − 9/30 = 11/30. Exam tip: For a gaining partner, subtract the old share from the new share.

B’s old share was \(1/(3+1+6)=1/10\). After C’s retirement, A and B share profits in the new ratio 2:1, so B’s new share is \(1/(2+1)=1/3\). Therefore, B’s gain is \(1/3-1/10=(10-3)/30=7/30\). Option 1/3 represents B’s new share, not the gain. Exam tip: Gain in share = New share − Old share.

Under the old ratio, A’s share was 3/10 and B’s share was 1/10. Under the new ratio of 2:1, A’s share becomes 2/3 and B’s share becomes 1/3. Therefore, A’s gain = 2/3 − 3/10 = 11/30, while B’s gain = 1/3 − 1/10 = 7/30. Hence, the gaining ratio is 11/30:7/30 = 11:7. Exam tip: Calculate each partner’s gain by subtracting the old share from the new share.