HCF of `(14)/(33),(21)/(22),(42)/(55)` is

Options

A. `(7)/(330)`
B. `(42)/(550)`
C. `(3)/(321)`
D. `(21)/(350)`

Answer

To find the HCF (Highest Common Factor) of the fractions \(\frac{14}{33}\), \(\frac{21}{22}\), and \(\frac{42}{55}\), we can follow these steps:

### Step 1: Identify the numerators and denominators
The fractions are:
- Numerators: 14, 21, 42
- Denominators: 33, 22, 55

### Step 2: Find the HCF of the numerators
To find the HCF of the numerators (14, 21, 42), we can list the factors:
- Factors of 14: 1, 2, 7, 14
- Factors of 21: 1, 3, 7, 21
- Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42

The common factors are 1 and 7. Thus, the HCF of the numerators is **7**.

### Step 3: Find the LCM of the denominators
Next, we find the LCM (Lowest Common Multiple) of the denominators (33, 22, 55):
- Prime factorization:
- 33 = 3 × 11
- 22 = 2 × 11
- 55 = 5 × 11

The LCM is found by taking the highest power of each prime:
- LCM = \(2^1 \times 3^1 \times 5^1 \times 11^1 = 330\)

### Step 4: Form the HCF of the fractions
Now that we have the HCF of the numerators (7) and the LCM of the denominators (330), we can form the HCF of the fractions:
\[ \text{HCF} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}} = \frac{7}{330} \]

### Final Answer
The HCF of the fractions \(\frac{14}{33}\), \(\frac{21}{22}\), and \(\frac{42}{55}\) is \(\frac{7}{330}\).

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