A and B share profits in the ratio 6:7. C is admitted for a \(\frac{4}{13}\) share of profits, and the remaining profit is divided between A and B in their old ratio. What will be the new profit-sharing ratio?

Correct answer: \(54:63:52\)

C’s share is \(\frac{4}{13}\), so the profit remaining for A and B is \(1-\frac{4}{13}=\frac{9}{13}\). A receives \(\frac{9}{13}\times\frac{6}{13}=\frac{54}{169}\), B receives \(\frac{9}{13}\times\frac{7}{13}=\frac{63}{169}\), and C receives \(\frac{4}{13}=\frac{52}{169}\). Therefore, the new ratio is \(54:63:52\). Option B is incorrect because \(4:13\) represents C’s share, not the new profit-sharing ratio. Exam tip: First subtract the new partner’s share from 1, then distribute the remainder in the old partners’ original ratio.