HCF of `6/5,3/25,9/15,12/5`

Options

A. `12/25`
B. `12/5`
C. `3/55`
D. `1/25`

Answer

To find the HCF (Highest Common Factor) of the fractions \( \frac{6}{5}, \frac{3}{25}, \frac{9}{15}, \frac{12}{5} \), we can follow these steps:

### Step 1: Identify the numerators and denominators
The fractions are:
- Numerators: 6, 3, 9, 12
- Denominators: 5, 25, 15, 5

### Step 2: Find the HCF of the numerators
To find the HCF of the numerators (6, 3, 9, 12):
- The factors of 6 are: 1, 2, 3, 6
- The factors of 3 are: 1, 3
- The factors of 9 are: 1, 3, 9
- The factors of 12 are: 1, 2, 3, 4, 6, 12

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

### Step 3: Find the LCM of the denominators
To find the LCM of the denominators (5, 25, 15, 5):
- The prime factorization of 5 is \( 5^1 \)
- The prime factorization of 25 is \( 5^2 \)
- The prime factorization of 15 is \( 3^1 \times 5^1 \)

The LCM is found by taking the highest power of each prime:
- For 5: \( 5^2 \)
- For 3: \( 3^1 \)

Thus, the LCM is \( 5^2 \times 3^1 = 25 \times 3 = 75 \).

### Step 4: Write the HCF of the fractions
Now that we have the HCF of the numerators (3) and the LCM of the denominators (75), we can write the HCF of the fractions:
\[ \text{HCF} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}} = \frac{3}{75} \]

### Step 5: Simplify the fraction
To simplify \( \frac{3}{75} \):
\[ \frac{3 \div 3}{75 \div 3} = \frac{1}{25} \]

### Final Answer
The HCF of the fractions \( \frac{6}{5}, \frac{3}{25}, \frac{9}{15}, \frac{12}{5} \) is \( \frac{1}{25} \).

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