Find the LCM `(1)/(2),(3)/(2) and (9)/(4)`.

Options

A. `(9)/(4)`
B. `(9)/(2)`
C. `(1)/(4)`
D. `(1)/(2)`

Answer

To find the LCM of the fractions \( \frac{1}{2}, \frac{3}{2}, \) and \( \frac{9}{4} \), we can follow these steps:

### Step 1: Identify the Numerators and Denominators
The fractions are:
- Numerators: \( 1, 3, 9 \)
- Denominators: \( 2, 2, 4 \)

### Step 2: Find the LCM of the Numerators
We need to find the LCM of the numerators \( 1, 3, \) and \( 9 \).

1. **List the multiples**:
- Multiples of \( 1 \): \( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, \ldots \)
- Multiples of \( 3 \): \( 3, 6, 9, 12, 15, \ldots \)
- Multiples of \( 9 \): \( 9, 18, 27, \ldots \)

2. **Identify the least common multiple**:
The smallest common multiple among \( 1, 3, \) and \( 9 \) is \( 9 \).

### Step 3: Find the HCF of the Denominators
Next, we find the HCF of the denominators \( 2, 2, \) and \( 4 \).

1. **List the factors**:
- Factors of \( 2 \): \( 1, 2 \)
- Factors of \( 4 \): \( 1, 2, 4 \)

2. **Identify the highest common factor**:
The highest common factor among \( 2, 2, \) and \( 4 \) is \( 2 \).

### Step 4: Calculate the LCM of the Fractions
The formula for the LCM of fractions is given by:

\[ \text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}} \]

Substituting the values we found:

\[ \text{LCM} = \frac{9}{2} \]

### Final Answer
Thus, the LCM of the fractions \( \frac{1}{2}, \frac{3}{2}, \) and \( \frac{9}{4} \) is \( \frac{9}{2} \).

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