https://file.aiquickdraw.com/mathbot/file/1726184370078eqjuxbvo.jpg

To solve this problem, we need to categorize each rational number in the "Answer Bank" into two categories: 

1. **Terminates in Zeros**
2. **Does Not Terminate in Zeros**

### List of Rational Numbers:
From the image, the numbers in the "Answer Bank" are:
- \(\frac{4}{15}\)
- \(28.178\)
- \(\frac{9}{5}\)
- \(0.2\)
- \(0.005\)

### Step-by-Step Analysis:

1. **Identify Terminating and Non-Terminating Decimals:**

   A decimal terminates if it ends after a finite number of digits. To determine if a fraction terminates, its denominator (in simplest form) should only have the prime factors \(2\) or \(5\). 

   - \(0.5\): Clearly terminates in zeros (as it can be expressed as \(\frac{1}{2}\)).
   - \(0.2\): Clearly terminates in zeros (as it can be expressed as \(\frac{1}{5}\)).
   - \(0.005\): Clearly terminates in zeros (as it can be expressed as \(\frac{1}{200}\)).
   
2. **Check the Fractions:**
   
   - **\(\frac{4}{15}\):**  
     The denominator \(15 = 3 \times 5\). Since it has a prime factor of \(3\), it does not terminate in zeros.
   
   - **\(\frac{9}{5}\):**  
     The denominator is \(5\), which is a prime factor. This will terminate in zeros when converted to decimal (\(9 \div 5 = 1.8\)).
   
3. **Check the Decimal Numbers:**

   - \(28.178\): Does not terminate in zeros; it has digits other than 0 after the decimal point.
   
### Final Categorization:

- **Terminates in Zeros:**
  - \(0.5\)
  - \(0.2\)
  - \(0.005\)
  - \(\frac{9}{5}\)

- **Does Not Terminate in Zeros:**
  - \(28.178\)
  - \(\frac{4}{15}\)

Would you like more details or explanations on any part of this process? Here are some related questions to think about:

1. What are other examples of fractions that terminate in zeros?
2. How do you convert repeating decimals into fractions?
3. What are the prime factors that affect whether a fraction terminates?
4. Why do some decimals have repeating sequences?
5. Can you explain the difference between rational and irrational numbers?

**Tip:** To determine if a fraction's decimal representation terminates, always check the factors of its simplified denominator!

Drag each rational number to the appropriate bin: 'Terminates in Zeros' or 'Does Not Terminate in Zeros'.

Learn how to classify rational numbers as terminating or non-terminating decimals using simple prime factorization methods. This math problem is ideal for students in grades 6-8.

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