New Profit Sharing Ratio

The remaining profit is (4/5) and sharing it in (3:2) gives A (12/25) and B (8/25). In exams first find the remaining share.

A and B sacrifice (1/8) each. New shares are (1/2,1/4,1/4) which convert to (4:2:2); check carefully before choosing. This row has an error.

The remaining share is (5/6). Dividing it in (2:1) gives (5/9,5/18,1/6) so the ratio is (10:5:3).

The ratio in which existing partners surrender their profit shares to admit a new partner is called the sacrificing ratio. Gaining ratio is generally relevant on retirement or death. Exam tip: sacrifice = old share − new share.

A's new share is (3/4-1/4=1/2) and B's share remains (1/4). So the ratio is (1/2:1/4:1/4=2:1:1).

The old ratio of 4:3 gives A a share of 4/7 and B a share of 3/7. Since C receives 2/7 entirely from B, B’s new share becomes 3/7 − 2/7 = 1/7, while A’s share remains 4/7. Therefore, A:B:C = 4/7:1/7:2/7 = 4:1:2. Exam tip: when the incoming partner’s share is acquired from only one existing partner, reduce only that partner’s share.

C's (1/9) share is taken in (2:1) so A sacrifices (2/27) and B sacrifices (1/27). New shares are (13/27,11/27,3/27).

The mathematically correct new ratio is 3:5:2, so option A is correct after correcting the original option text. C receives 1/5, and the sacrifice is equal, so A and B each give up half of 1/5, which is 1/10. A’s old share is 2/5=4/10; after sacrifice it becomes 4/10−1/10=3/10. B’s old share is 3/5=6/10; after sacrifice it becomes 6/10−1/10=5/10. C receives 1/5=2/10. Therefore the new shares are 3/10, 5/10, and 2/10, giving 3:5:2. Their total is 3+5+2=10, so the shares add to the whole profit. Option B leaves the old shares unchanged and gives C an incorrectly scaled share. Option C does not subtract equal sacrifices correctly, and option D reverses the old partners’ relative shares. The key warning is that an equal sacrifice means equal fractions are subtracted, not equal proportions of the old shares.

A's new share is (7/10-1/20=13/20) and B's is (3/10-1/20=5/20). C's share is (2/20) so the ratio is (13:5:2).

A’s old share is 3/5 and B’s is 2/5. After giving up 1/10 each, their new shares are A = 3/5 − 1/10 = 1/2 and B = 2/5 − 1/10 = 3/10. C receives 1/10 + 1/10 = 1/5. Thus, the new ratio is 1/2 : 3/10 : 1/5 = 5:3:2.

After C receives 1/5 of the profit, the remaining profit is 1 − 1/5 = 4/5. A’s old share was 6/(6+4) = 3/5. Therefore, A’s new share is \(\frac{4}{5}\times\frac{3}{5}=\frac{12}{25}\). Option B is only A’s old share and does not account for the reduction due to C’s admission. Exam tip: When the remaining profit is shared in the old ratio, each old partner’s new share equals the remaining profit multiplied by that partner’s old fraction.

C receives \(1/5\) of the total profit, so the portion left for A and B is \(1-1/5=4/5\). B’s old share was \(7/(8+7)=7/15\). Therefore, B’s new share is \(\frac{4}{5}\times\frac{7}{15}=\frac{28}{75}\). The value \(7/15\) represents B’s old share before C’s admission, not the new share. Exam tip: First calculate the profit left for the old partners, then multiply it by the concerned partner’s old fraction.

C receives 1/10 of the total profit, so the balance available to A and B is 9/10. This balance is divided in the old ratio 4:1: A receives 9/10 × 4/5 = 18/25 = 36/50, while B receives 9/10 × 1/5 = 9/50. C’s share is 1/10 = 5/50. Therefore, the new ratio is 36:9:5. Exam tip: first calculate the balance left for the old partners, then distribute it in their old ratio.

Direct answer: Option A, 6:4:5, is correct. The old ratio of A and B is 9:6, which simplifies to 3:2. C receives 1/3, so only 2/3 remains for A and B. That remaining 2/3 must be divided in the old ratio 3:2. A receives (2/3)×(3/5)=2/5, and B receives (2/3)×(2/5)=4/15. C receives 1/3 = 5/15. Expressing every share with denominator 15 gives A 6/15, B 4/15 and C 5/15, hence 6:4:5. Option A is therefore correct. Option B, 9:6:3, treats the original figures as if they could simply be joined with C’s share; their total does not represent the required normalized distribution. Option C gives C three parts instead of five in the common denominator. Option D does not preserve the correct remaining-profit calculation. Exam cue: when the new partner gets a stated share and the remainder follows the old ratio, multiply the remainder by each old partner’s fraction.

C receives 1/4 of the total profit, so 3/4 remains for A and B. Since their old ratio is 11:9, its total is 20. Therefore, A’s new share is 3/4 × 11/20 = 33/80, B’s is 3/4 × 9/20 = 27/80, and C’s share is 1/4 = 20/80. Hence, the new ratio is 33:27:20. Option B is incorrect because it treats C’s share as though it could simply be added to the old ratio. Exam tip: Multiply each old partner’s fraction by the profit remaining after the new partner’s share is deducted.

When the new share is taken in old ratio the old partners share the balance in old ratio. Therefore the ratio is (30:12:7).

Let the total profit be 1. C receives 1/4, so the amount left for A and B is 1 − 1/4 = 3/4. The question specifically states that A and B divide this remainder equally, so each receives (3/4) ÷ 2 = 3/8. C’s share, 1/4, is equal to 2/8. Thus the three shares are 3/8, 3/8 and 2/8, which produce the new ratio 3:3:2. Therefore option A is correct. The ratio sums to 8, confirming the shares total one whole. Option B would give C only one-third, while options C and D do not preserve C’s fixed one-fourth share together with equal distribution of the remainder.

A's new share is (3/5-1/15=8/15) and B's is (2/5-1/30=11/30). With common denominator the ratio is (16:11:3) not option A.

C receives 1/4 of the total profit, leaving 3/4 for A and B. Dividing this remainder in the old ratio 3:5 gives A's share as (3/4) × (3/8) = 9/32 and B's share as (3/4) × (5/8) = 15/32. C's share is 1/4 = 8/32. Therefore, the new profit-sharing ratio is 9:15:8. Option B incorrectly keeps A and B's old ratio without adjusting it for the remaining profit. Exam tip: First find the profit left for the old partners, then distribute it in their old ratio.

C receives 2/9 of the total profit, so the remaining share for A and B is 7/9. Since their old ratio 4:5 has a total of 9, A receives \(\frac{7}{9}\times\frac{4}{9}=\frac{28}{81}\) and B receives \(\frac{7}{9}\times\frac{5}{9}=\frac{35}{81}\). C’s share is \(\frac{2}{9}=\frac{18}{81}\). Therefore, the new ratio is 28:35:18. Exam tip: First calculate the remaining share, then distribute it among the old partners in their old ratio.

Convert A and B's old ratio into fractions: A=5/12 and B=7/12. Since C takes 1/6=2/12 from A, A's remaining share is 5/12−2/12=3/12=1/4. B remains 7/12 and C receives 1/6=2/12. Expressing all shares over 12 gives 3:7:2. Therefore option A is correct; the other ratios either fail to deduct C's share correctly or use inconsistent proportions.

The old ratio is 6:5, so A’s and B’s old shares are 6/11 and 5/11, respectively. Since C receives 2/11 entirely from B, B’s new share becomes 5/11 − 2/11 = 3/11, while A’s share remains 6/11. Therefore, A:B:C = 6/11:3/11:2/11 = 6:3:2. Exam tip: If the incoming partner acquires the share entirely from one existing partner, reduce only that partner’s share.

C's (1/5) share is taken in (3:1) so A sacrifices (3/20) and B sacrifices (1/20). New shares are (17/60,31/60,12/60).

Direct answer: Option A, 7:3:3, is correct. The old ratio A:B is 9:4, so their old shares are 9/13 and 4/13. C receives 3/13. The question says this is sacrificed by A and B in the ratio 2:1. Since the total sacrifice is 3/13, A sacrifices 2/13 and B sacrifices 1/13. A’s new share is 9/13 − 2/13 = 7/13. B’s new share is 4/13 − 1/13 = 3/13. C’s share remains 3/13. Removing the common denominator 13 gives 7:3:3. Option A matches this calculation. Option B simply repeats the old ratio and adds C without reducing A and B. Option C does not use either the old ratio or the stated sacrifice ratio. Option D doubles A’s and B’s old parts and is not a normalized ratio. Always divide a new partner’s share among sacrificing partners before subtracting.

A’s old share is 3/(3+4) = 3/7. Since C’s 1/7 share is taken from A and B in the ratio 1:2, A’s sacrifice is 1/7 × 1/3 = 1/21. Therefore, A’s new share is 3/7 − 1/21 = 9/21 − 1/21 = 8/21. Exam tip: A partner’s new share equals the old share minus the share sacrificed. Option B is only A’s old share and ignores the sacrifice.