Let the total runs scored in the first 30 matches be $x$.

Let the number of additional matches to be played be $n$.

Case 1: Average is 20 after scoring 480 runs

If the cricketer scores another 480 runs, his total runs become $x + 480$. The total number of matches will then be $30 + n$. The average score is given as 20:

$\frac{x + 480}{30 + n} = 20$

Multiply through by $30 + n$:

$x + 480 = 20(30 + n)$

$x + 480 = 600 + 20n$

$x = 600 + 20n - 480$

x = 120 + 20n \tag{1}

Case 2: Average is 24 after scoring 600 runs

If the cricketer scores another 600 runs, his total runs become $x + 600$. The total number of matches will be $30 + n$. The average score is given as 24:

$\frac{x + 600}{30 + n} = 24$

Multiply through by $30 + n$:

$x + 600 = 24(30 + n)$

$x + 600 = 720 + 24n$

$x = 720 + 24n - 600$

x = 120 + 24n \tag{2}

Solving for $n$:

From equations $(1)$ and $(2)$:

$120 + 20n = 120 + 24n$

$20n = 24n$

$4n = 0$

$n = 5$

Finding $x$:

Substitute $n = 5$ into $x = 120 + 20n$:

$x = 120 + 20(5)$

$x = 120 + 100$

$x = 220$

Conclusion:

The average runs scored in the first 30 matches is:

$\frac{x}{30} = \frac{220}{30} = 7.33 \, (\text{rounded to 2 decimal places}).$

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Do you have further questions or want detailed clarifications? Here are 5 related questions for exploration:

1. How can we verify the solution using both conditions simultaneously?
2. What is the total number of matches played in this year?
3. How does the cricketer’s performance in future matches affect his average score?
4. What would the average be if the cricketer scores another 720 runs instead?
5. How can this problem be solved graphically to visualize $x$ and $n$?

Tip: When solving problems involving averages, set up equations systematically to account for all variables.