X, Y and Z are partners in the ratio $$\frac{1}{2}:\frac{2}{5}$$ and $$\frac{1}{{10}}$$. If X retires then what is the profit sharing ratio between remaining partners?

The correct answer is $\boxed{\text{C. }5:1}$.

The initial profit sharing ratio between X, Y and Z is $\frac{1}{2}:\frac{2}{5}:\frac{1}{{10}}$. This can be written as $10:12:6$. If X retires, the new profit sharing ratio between Y and Z is $12:6$, which is equal to $5:1$.

Here is a brief explanation of each option:

• Option A: $2:1$. This is not the correct answer because the initial profit sharing ratio between X, Y and Z is not $1:2$.
• Option B: $4:1$. This is not the correct answer because the initial profit sharing ratio between X, Y and Z is not $2:1$.
• Option C: $5:1$. This is the correct answer because the new profit sharing ratio between Y and Z is $5:1$.
• Option D: $3:1$. This is not the correct answer because the new profit sharing ratio between Y and Z is not $3:1$.