A bag contains a certain number of black and white balls, of which 90% are black. When 9 white balls are added to the bag, the ratio of the black balls to the white balls is $$4 \colon 3$$. The number of white balls in the bag at the beginning is

Let the total number of balls initially be $$x$$.

Since $$90\%$$ of the balls are black,

$$\text{Number of black balls}=90\% \text{ of }x=\frac{90}{100}x=\frac{9x}{10}.$$

Therefore, the number of white balls initially is

$$x-\frac{9x}{10}=\frac{x}{10}.$$

When 9 white balls are added,

$$\text{Number of black balls}=\frac{9x}{10}$$

and

$$\text{Number of white balls}=\frac{x}{10}+9.$$

According to the question, the ratio of black balls to white balls is $$4:3$$.

Therefore,

$$\frac{\frac{9x}{10}}{\frac{x}{10}+9}=\frac{4}{3}.$$

Cross-multiplying,

$$3\left(\frac{9x}{10}\right)= 4\left(\frac{x}{10}+9\right).$$

$$\frac{27x}{10} = \frac{4x}{10}+36.$$

Multiplying throughout by $$10$$,

$$27x=4x+360.$$

$$23x=360.$$

$$x=\frac{360}{23}.$$

But the number of balls must be a whole number. Hence, the given data are inconsistent.

Therefore, $$\textbf{there is no possible whole-number solution for the number of white balls initially}.$$

$${\text{No valid whole-number solution}}$$