X, Y and Z are partners in the ratio of $\frac{1}{2}:\frac{2}{5}:\frac{1}{10}$. What will be the new ratio of the remaining partner if X retires?

The old ratio of X, Y and Z is $\frac{5}{10}:\frac{4}{10}:\frac{1}{10}=5:4:1$.If X retires, the remaining partners Y and Z will share profits in the ratio of their old shares, which is $4:1$.