Without actual division, find which of the following rational numbers have terminating decimal representation :
`(i)(5)/(32)," "(ii)(3)/(320)," "(iii)(7)/(24)`

Explanation

To determine which of the given rational numbers have a terminating decimal representation, we need to check the form of their denominators. A rational number has a terminating decimal representation if its denominator (in simplest form) can be expressed as \(2^m \times 5^n\), where \(m\) and \(n\) are non-negative integers.

Let's analyze each of the given rational numbers step by step:

### Step 1: Analyze \( \frac{5}{32} \)

1. **Identify the denominator**: The denominator is \(32\).
2. **Factor the denominator**:
\[ 32 = 2^5 \]
3. **Check the form**: The denominator can be expressed as \(2^m\) where \(m = 5\) and \(5^0\) (since \(5\) is not present).
4. **Conclusion**: Since the denominator is of the form \(2^m \times 5^n\), \( \frac{5}{32} \) has a terminating decimal representation.

### Step 2: Analyze \( \frac{3}{320} \)

1. **Identify the denominator**: The denominator is \(320\).
2. **Factor the denominator**:
\[ 320 = 32 \times 10 = 2^5 \times (2 \times 5) = 2^6 \times 5^1 \]
3. **Check the form**: The denominator can be expressed as \(2^m \times 5^n\) where \(m = 6\) and \(n = 1\).
4. **Conclusion**: Since the denominator is of the form \(2^m \times 5^n\), \( \frac{3}{320} \) has a terminating decimal representation.

### Step 3: Analyze \( \frac{7}{24} \)

1. **Identify the denominator**: The denominator is \(24\).
2. **Factor the denominator**:
\[ 24 = 8 \times 3 = 2^3 \times 3^1 \]
3. **Check the form**: The denominator contains \(3\), which is not of the form \(2^m \times 5^n\).
4. **Conclusion**: Since the denominator includes a factor of \(3\), \( \frac{7}{24} \) does not have a terminating decimal representation.

### Final Results

- \( \frac{5}{32} \) - Terminating
- \( \frac{3}{320} \) - Terminating
- \( \frac{7}{24} \) - Non-terminating