A and B share profits in the ratio 2:3. C is admitted as a partner for a \(\frac{1}{4}\) share of profits. If the new mutual ratio of A and B is 3:2, which partner will gain?

Correct answer: A gains \(\frac{1}{20}\)

After C receives \(\frac{1}{4}\) of the profits, \(\frac{3}{4}\) remains for A and B. Under their new mutual ratio of 3:2, A’s new share is \(\frac{3}{4}\times\frac{3}{5}=\frac{9}{20}\), whereas A’s old share was \(\frac{2}{5}=\frac{8}{20}\). Therefore, A gains \(\frac{9}{20}-\frac{8}{20}=\frac{1}{20}\). B’s new share is \(\frac{3}{4}\times\frac{2}{5}=\frac{3}{10}\), which is less than B’s old share of \(\frac{3}{5}\), so B sacrifices. Exam tip: First subtract the incoming partner’s share from 1, then distribute the remainder according to the continuing partners’ new mutual ratio.