Which of the following rational numbers have non-terminating decimal expansion?

Options

A. `144/225`
B. `25/36`
C. `49/256`
D. `7/250`

Answer

To determine which of the given rational numbers have non-terminating decimal expansions, we need to understand the conditions for a rational number to have a terminating decimal expansion. A rational number in the form \( \frac{p}{q} \) has a terminating decimal expansion if the denominator \( q \) (in its simplest form) can be expressed as \( 2^m \times 5^n \), where \( m \) and \( n \) are non-negative integers.

Let's analyze each option step by step:

### Step 1: Analyze the first option \( \frac{144}{225} \)

1. **Factor the denominator**:
- \( 225 = 15^2 = (3 \times 5)^2 = 3^2 \times 5^2 \)
2. **Check the form**:
- The denominator has a factor of \( 3 \), which means it cannot be expressed in the form \( 2^m \times 5^n \).
3. **Conclusion**:
- This number has a **non-terminating decimal expansion**.

### Step 2: Analyze the second option \( \frac{25}{36} \)

1. **Factor the denominator**:
- \( 36 = 6^2 = (2 \times 3)^2 = 2^2 \times 3^2 \)
2. **Check the form**:
- The denominator has a factor of \( 3 \), which means it cannot be expressed in the form \( 2^m \times 5^n \).
3. **Conclusion**:
- This number has a **non-terminating decimal expansion**.

### Step 3: Analyze the third option \( \frac{49}{256} \)

1. **Factor the denominator**:
- \( 256 = 2^8 \)
2. **Check the form**:
- The denominator can be expressed as \( 2^8 \times 5^0 \), which fits the required form.
3. **Conclusion**:
- This number has a **terminating decimal expansion**.

### Step 4: Analyze the fourth option \( \frac{7}{250} \)

1. **Factor the denominator**:
- \( 250 = 25 \times 10 = 5^3 \times 2^1 \)
2. **Check the form**:
- The denominator can be expressed as \( 2^1 \times 5^3 \), which fits the required form.
3. **Conclusion**:
- This number has a **terminating decimal expansion**.

### Final Conclusion

The rational numbers with non-terminating decimal expansions from the options are:

- Option 1: \( \frac{144}{225} \)
- Option 2: \( \frac{25}{36} \)