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?

Step 1: Let the current age of Antony be $8x$ and Betty be $9x$.
Step 2: Four years from now, their ages will be $(8x+4)$ and $(9x+4)$ respectively. According to the question, this new ratio is $9:10$:

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

Step 3: Cross-multiply to solve for $x$:

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

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

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

$$x=4$$

Step 4: Compute their current ages using the value of $x$:

* $\text{Current age of Antony}=8x=8\times 4=32\text{ years}$
* $\text{Current age of Betty}=9x=9\times 4=36\text{ years}$