The old ratio is (6:4:3). The new partner receives (1/13) share from the second partner and (2/13) share from the third partner. What is the new ratio?

Correct answer: (6:3:1:3)

The old ratio 6:4:3 has a total of 13, so the old shares are 6/13 for the first partner, 4/13 for the second, and 3/13 for the third. The new partner receives 1/13 from the second partner and 2/13 from the third partner. The first partner gives nothing and therefore remains at 6/13. The second partner gives 1/13, so the new share is 4/13 − 1/13 = 3/13. The third gives 2/13, so the new share is 3/13 − 2/13 = 1/13. The incoming partner receives 1/13 + 2/13 = 3/13. Therefore the new ratio, in the order first, second, third, and new partner, is 6:3:1:3, so option A is correct.

A useful check is that the new parts add to 6 + 3 + 1 + 3 = 13, matching the whole profit. The first partner’s share is unchanged, even though the total number of partners increases. The new partner’s share must be taken from the existing partners, and the stated contributions show exactly who gives up the shares. Hence choices that leave old shares unchanged or subtract from the wrong partners cannot be correct.