HCF of 12/5, 21/6, 9/4, and 18/3 is:

Options

A. `1//2`
B. `3//2`
C. `3//5`
D. `3//7`

Answer

To find the HCF (Highest Common Factor) of the fractions \( \frac{12}{5}, \frac{21}{6}, \frac{9}{4}, \frac{18}{3} \), we will follow these steps:

### Step 1: Identify the Numerators and Denominators
The fractions are:
- \( \frac{12}{5} \) (Numerator: 12, Denominator: 5)
- \( \frac{21}{6} \) (Numerator: 21, Denominator: 6)
- \( \frac{9}{4} \) (Numerator: 9, Denominator: 4)
- \( \frac{18}{3} \) (Numerator: 18, Denominator: 3)

### Step 2: Calculate the HCF of the Numerators
We need to find the HCF of the numerators: 12, 21, 9, and 18.

- **Factors of 12**: 1, 2, 3, 4, 6, 12
- **Factors of 21**: 1, 3, 7, 21
- **Factors of 9**: 1, 3, 9
- **Factors of 18**: 1, 2, 3, 6, 9, 18

The common factors are: 1, 3. The highest common factor is **3**.

### Step 3: Calculate the LCM of the Denominators
Next, we find the LCM of the denominators: 5, 6, 4, and 3.

- **Multiples of 5**: 5, 10, 15, 20, 25, 30, ...
- **Multiples of 6**: 6, 12, 18, 24, 30, ...
- **Multiples of 4**: 4, 8, 12, 16, 20, 24, 28, 32, ...
- **Multiples of 3**: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...

The smallest common multiple is **60**.

### Step 4: Form the HCF of the Fractions
Now we can form the HCF of the fractions using the HCF of the numerators and the LCM of the denominators.

\[ \text{HCF} = \frac{\text{HCF of Numerators}}{\text{LCM of Denominators}} = \frac{3}{60} \]

### Step 5: Simplify the Fraction
Now, we simplify \( \frac{3}{60} \):

\[ \frac{3}{60} = \frac{1}{20} \]

### Final Answer
The HCF of \( \frac{12}{5}, \frac{21}{6}, \frac{9}{4}, \frac{18}{3} \) is \( \frac{1}{20} \).

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