Which of the following numbers have terminating decimal expansion?

Options

A. `(8)/(225)`
B. `(5)/(18)`
C. `(11)/(21)`
D. `(21)/(150)`

Answer

To determine which of the given fractions has a terminating decimal expansion, we need to analyze each option based on the properties of terminating decimals. A fraction in the form of \( \frac{a}{b} \) has a terminating decimal expansion if the prime factorization of the denominator \( b \) (after simplifying the fraction) contains only the prime factors 2 and/or 5.

Let's analyze each option step by step:

### Step 1: Analyze Option A - \( \frac{8}{225} \)

1. **Factor the denominator**:
- \( 225 = 15 \times 15 = 3^2 \times 5^2 \)
2. **Check the prime factors**:
- The denominator has a factor of 3, which is not 2 or 5.
3. **Conclusion**:
- This fraction does not have a terminating decimal expansion.

### Step 2: Analyze Option B - \( \frac{5}{18} \)

1. **Factor the denominator**:
- \( 18 = 2 \times 9 = 2 \times 3^2 \)
2. **Check the prime factors**:
- The denominator has a factor of 3, which is not 2 or 5.
3. **Conclusion**:
- This fraction does not have a terminating decimal expansion.

### Step 3: Analyze Option C - \( \frac{11}{21} \)

1. **Factor the denominator**:
- \( 21 = 3 \times 7 \)
2. **Check the prime factors**:
- The denominator has factors of 3 and 7, neither of which are 2 or 5.
3. **Conclusion**:
- This fraction does not have a terminating decimal expansion.

### Step 4: Analyze Option D - \( \frac{21}{150} \)

1. **Factor the denominator**:
- \( 150 = 2 \times 75 = 2 \times 3 \times 5^2 \)
2. **Check the prime factors**:
- The denominator has a factor of 2 and a factor of 5, but also a factor of 3.
3. **Conclusion**:
- This fraction does not have a terminating decimal expansion.

### Final Conclusion

After analyzing all options, we find that none of the options have a terminating decimal expansion. Therefore, the answer is that none of the given fractions \( \frac{8}{225}, \frac{5}{18}, \frac{11}{21}, \frac{21}{150} \) have a terminating decimal expansion.