What are the current ages of Antony and Betty if their ages are in the ratio $8:9$ now and will be in the ratio $9:10$ four years from now?

Let the current ages of Antony and Betty be $8x$ and $9x$ respectively.
According to the question, four years from now, the ratio of their ages will be $9:10$:

$$\frac{8x+4}{9x+4}=\frac{9}{10}$$

Cross-multiplying to solve for $x$:

$$10(8x+4)=9(9x+4)$$

$$80x+40=81x+36$$

$$40-36=81x-80x$$

$$x=4$$

Now, calculate their current ages:

* Antony's current age = $8x=8\times 4=32$ years
* Betty's current age = $9x=9\times 4=36$ years