Explanation:

To solve this problem, let's break it down into steps:

- The average age of $x$ students is 20. Therefore, the total age of these students is $20x$.

- Four new students are admitted, each with an age of 7.5. So, the total additional age is $4\times 7.5=30$.

- The new average age decreases by 5, resulting in an average of $20-5=15$.

- The total number of students now is $x+4$.

- The equation for the new average is:

$$

\frac{20x + 30}{x + 4} = 15

$$

- Simplifying this, we have:

$$

20x + 30 = 15(x + 4)

$$

$$

20x + 30 = 15x + 60

$$

$$

5x = 30

$$

$$

x = 6

$$

- Therefore, the value of $x$ is 6.

- ?? Option 4: 6 is the correct answer.