New Ratios

For each continuing partner, calculate the change as: new share − old share. A positive difference represents a gain, whereas a negative difference represents a sacrifice; therefore, a negative difference must not be included in the gaining ratio. Option B is incorrect because converting a negative difference into a positive one would ignore the partner’s actual sacrifice. Exam tip: calculate the difference for every partner separately, then form the ratio using only the positive differences.

A’s old share was \(\frac{5}{5+2+3}=\frac{5}{10}\). Under the new ratio of 7:3, A’s new share is \(\frac{7}{10}\). Therefore, A’s gain is \(\frac{7}{10}-\frac{5}{10}=\frac{2}{10}=\frac{1}{5}\). Exam tip: Gain of a partner = New share − Old share.

B’s old share was 2/10. In the new ratio of 7:3, B’s share is 3/10. Therefore, B’s gaining share = new share − old share = 3/10 − 2/10 = 1/10. Exam tip: calculate a partner’s gain by subtracting the old share from the new share.

Under the old ratio, A's share was 5/10 and B's share was 2/10. Under the new ratio, A receives 7/10 and B receives 3/10. Therefore, A's gain is 7/10 − 5/10 = 1/5, while B's gain is 3/10 − 2/10 = 1/10. Hence, the gaining ratio is 1/5 : 1/10 = 2:1. The ratio 7:3 is the new ratio, not the gaining ratio. Exam tip: Gaining ratio = New ratio − Old ratio.

After C’s retirement, A and B share profits in the new ratio 2:3. The total parts are 2+3 = 5, so A’s new share is 2/5. Option C, 3/5, is B’s new share, while 1/10 was A’s old share; therefore, the old ratio should not be used for this calculation. Exam tip: Divide the partner’s part in the new ratio by the total parts of that ratio.

The correct answer is Option A, 3:4. The old ratio A:B:C is 1:2:7, so A's old share is 1/10 and B's old share is 2/10. C retires. The new ratio of A and B is 2:3, so their new shares are 2/5 and 3/5. Calculate each gain separately. A's gain is 2/5 - 1/10 = 4/10 - 1/10 = 3/10. B's gain is 3/5 - 2/10 = 6/10 - 2/10 = 4/10. Therefore the gaining ratio is 3/10:4/10 = 3:4. Option A is correct. Option B, 2:3, is the new total profit-sharing ratio, not the gain ratio. Option C, 1:2, incorrectly compares the old shares or otherwise ignores the actual increases. Option D, 1:1, would mean both partners gained equally, but B gained 4/10 while A gained 3/10. The reliable method is always: new share minus old share for each continuing partner, followed by simplification. Memory cue: gain is not the new share; gain is only the increase.

D share is 4/10 and in 1:1:2 A gets 1/10 B gets 1/10 and C gets 2/10. New shares are 4/10 2/10 and 4/10.

The total ratio is 10, so D’s share is 4/10. Distributing this share among A, B and C in the ratio 1:1:2 gives them additional shares of 1/10, 1/10 and 2/10, respectively. Thus, their new shares are A = 4/10, B = 2/10 and C = 4/10. Therefore, the new ratio is 4:2:4, which simplifies to 2:1:2. The ratio 3:1:2 is the old ratio, while 1:1:2 is only the ratio for taking over D’s share. Exam tip: First calculate the retiring partner’s share, then distribute it according to the stated gain ratio.

A's old share was 2/(2+6+2) = 2/10 = 1/5. Under the new ratio 3:2, A's new share is 3/5. Therefore, A's gain = new share − old share = 3/5 − 1/5 = 2/5. Option B is A's new share, not the gain. Exam tip: Always subtract the old share from the new share to calculate a partner's gain.

C’s old share was 2/(2+6+2) = 2/10 = 1/5. Under the new ratio of 3:2, C’s new share is 2/5. Therefore, C’s gain = new share − old share = 2/5 − 1/5 = 1/5. The fraction 2/5 is C’s new share, not the gain. Exam tip: Always subtract the old share from the new share to calculate gain.

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

A’s original share is 1/10. This share is acquired by B, C and D in the ratio 1:2:2, so their respective gains are 1/50, 2/50 and 2/50. Therefore, B’s new share is 2/10 + 1/50 = 11/50, C’s is 3/10 + 2/50 = 17/50, and D’s is 4/10 + 2/50 = 22/50. Hence, the new ratio is 11:17:22. Exam tip: Divide the deceased partner’s share according to the gaining ratio and add each gain to the partner’s old share.

A’s old share was \(3/10\), and the new share is \(5/10\); therefore, A gains \(2/10\). Similarly, B also gains \(2/10\), since its share rises from \(3/10\) to \(5/10\). Hence, the gaining ratio is \(2/10:2/10=1:1\). Exam tip: calculate each continuing partner’s gain as new share minus old share; do not treat the new ratio itself as the gaining ratio.

B’s old share was 4/(3+4+3) = 4/10. After A’s retirement, B’s new share is 7/(7+5) = 7/12. Therefore, B’s gain = New share − Old share = 7/12 − 4/10 = 35/60 − 24/60 = 11/60. Option B is only B’s new share, not the gain. Exam tip: Calculate a partner’s gain by subtracting the old share from the new share.

C's old share was \(3/(3+4+3)=3/10\). Under the new ratio 7:5, C's new share is \(5/(7+5)=5/12\). Therefore, C's gain = new share − old share = \(5/12-3/10=(25-18)/60=7/60\). Exam tip: gain is calculated as the excess of the new share over the old share.

Under the old ratio, B’s share was 4/10 = 24/60 and C’s share was 3/10 = 18/60. Under the new ratio of 7:5, B’s share becomes 7/12 = 35/60 and C’s share becomes 5/12 = 25/60. Therefore, B’s gain is 35/60 − 24/60 = 11/60, while C’s gain is 25/60 − 18/60 = 7/60. Hence, the gaining ratio is 11:7. Exam tip: Calculate each partner’s gain as new share minus old share, then compare the gains.

B share increased from 3/10 to 5/10 and C share remained 5/10. Therefore only B gained A share.

Under the old ratio 2:3:5, B and C had shares of 3/10 and 5/10 respectively. The new ratio 5:5 means each receives 1/2, or 5/10. Therefore, B’s gain is 5/10 − 3/10 = 2/10, while C’s gain is 5/10 − 5/10 = 0. Hence, the gaining ratio is 2/10:0 = 1:0. Exam tip: Calculate each partner’s gain by subtracting the old share from the new share.

C share is 3/10. Dividing it in 2:2:1 gives new shares of A B D as 26/50 16/50 and 8/50 so the simplified ratio is 13:8:4.

If no information is given the old mutual ratio of continuing partners becomes the new ratio. So A B and D ratio will be 3:2:1.

When the share is acquired in old mutual ratio the ratio of continuing partners does not change. A and C old mutual ratio is 7:4.

C share is 2/10 and equal acquisition gives 1/10 each to A and B. New shares are 6/10 and 4/10 so the ratio is 3:2.

A’s profit share is 6/10. B acquires 1/3 of A’s share, which is 2/10, while C acquires 2/3, which is 4/10. Therefore, B’s new share is 2/10 + 2/10 = 4/10, and C’s new share is 2/10 + 4/10 = 6/10. Thus, the new ratio is 4:6 = 2:3. The ratio 1:2 is only the proportion in which A’s share is acquired, not the final ratio of B and C.

B’s original share is 6/10 = 12/20. In the 3:1 acquisition ratio, A receives 3/4 of B’s share, or 9/20, while C receives 1/4, or 3/20. A’s new share is 4/20 + 9/20 = 13/20, and C’s new share is 4/20 + 3/20 = 7/20. Therefore, the new ratio is 13:7. Remember that 3:1 is the gaining ratio for B’s share, not the final profit-sharing ratio.

A’s old share was 7/(7+2+1) = 7/10. Under the new ratio 3:2, A’s new share is 3/5 = 6/10. Thus, A’s share decreases from 7/10 to 6/10, so A sacrifices 1/10. Option B is incorrect because A’s share has decreased, not increased. Exam tip: Subtract the old share from the new share; a negative result indicates sacrifice, while a positive result indicates gain.