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!