To determine the next number in the series $45, 15, 30, 30, 10, 20$, let's analyze the pattern:

• Step 1: The first term is $45$.
• Step 2: The second term $15$ is obtained by dividing $45$ by 3.
• Step 3: The third term $30$ is obtained by multiplying $15$ by 2.
• Step 4: The fourth term $30$ remains the same.
• Step 5: The fifth term $10$ is obtained by dividing $30$ by 3.
• Step 6: The sixth term $20$ is obtained by multiplying $10$ by 2.

Following this observed sequence:

• We divide by 3, multiply by 2, repeat the same term, divide by 3, and multiply by 2.

To continue:

• Next Step: We divide $20$ by 3, which gives approximately $6.67$.

So, the next term in the series would likely be:

$6.67$

Would you like further details, or have any questions?

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Here are some additional questions you may find interesting:

1. What if we multiplied or divided differently in the sequence?
2. Can we form a similar sequence with different initial values?
3. How would the sequence change if we added a repeating addition or subtraction step?
4. How might rounding influence sequences with non-integer values?
5. What are other methods to predict patterns in sequences?

Tip: For number sequences, check for common operations like addition, subtraction, multiplication, and division.