Problem 57

To solve this problem, we'll break it down into two parts as given in the problem statement.

### Part 1: Finding the Average Percentage Reduction

Let's denote the average percentage reduction in price each time by $x$. This means that after each reduction, the price retains $(1-x)$ of its previous value. Since there are two consecutive reductions, the final price can be represented as the initial price multiplied by $(1-x)^2$. Given that the initial price is $30$ yuan per kilogram and the final price after two reductions is $19.2$ yuan per kilogram, we can set up the equation as follows:

$$30(1-x)^2 = 19.2$$

To solve for $x$, we first divide both sides by $30$:

$$(1-x)^2 = \frac{19.2}{30}$$

Simplifying the right side gives us:

$$(1-x)^2 = 0.64$$

Taking the square root of both sides, we find:

$$1-x = \sqrt{0.64} = 0.8$$

Solving for $x$ gives us two potential solutions since the square root operation can yield both a positive and a negative result. However, in the context of a percentage reduction, only the positive result makes sense:

$$x = 1 - 0.8 = 0.2$$

Converting $x$ from a decimal to a percentage, we get:

$$x = 20\%$$

Thus, the average percentage reduction in price each time is $\boxed{20\%}$.

### Part 2: Calculating the Selling Price After the Next Reduction

Now, using the average percentage reduction found in part 1, we calculate the selling price after the next price reduction. The current selling price is $19.2$ yuan per kilogram, and we reduce this price by $20\%$:

$$19.2 \times (1-20\%) = 19.2 \times 0.8 = 15.36$$

Therefore, the selling price after the next price reduction is $\boxed{15.36 \text{ yuan per kilogram}}$.