The sum of HCF and LCM of `(2)/(3),(4)/(9) and (5)/(6)` is

Options

A. `(111)/(9)`
B. `(31)/(5)`
C. `(121)/(8)`
D. `(35)/(6)`

Answer

To find the sum of the HCF and LCM of the fractions \( \frac{2}{3}, \frac{4}{9}, \frac{5}{6} \), we will follow these steps:

### Step 1: Identify the Numerators and Denominators
The fractions given are:
- Numerators: 2, 4, 5
- Denominators: 3, 9, 6

### Step 2: Find the HCF of the Numerators
To find the HCF (Highest Common Factor) of the numerators (2, 4, 5):
- The factors of 2: 1, 2
- The factors of 4: 1, 2, 4
- The factors of 5: 1, 5

The only common factor is 1. Therefore,
\[ \text{HCF of numerators} = 1 \]

### Step 3: Find the LCM of the Numerators
To find the LCM (Lowest Common Multiple) of the numerators (2, 4, 5):
- The multiples of 2: 2, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 25, 26, 28, 30, ...
- The multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
- The multiples of 5: 5, 10, 15, 20, 25, 30, 35, ...

The LCM is found by taking the highest power of each prime factor:
- For 2: \( 2^2 \) (from 4)
- For 5: \( 5^1 \)

Thus,
\[ \text{LCM of numerators} = 2^2 \times 5 = 4 \times 5 = 20 \]

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

The common factor is 3. Therefore,
\[ \text{HCF of denominators} = 3 \]

### Step 5: Find the LCM of the Denominators
To find the LCM of the denominators (3, 9, 6):
- The multiples of 3: 3, 6, 9, 12, 15, 18, ...
- The multiples of 9: 9, 18, 27, ...
- The multiples of 6: 6, 12, 18, ...

The LCM is found by taking the highest power of each prime factor:
- For 3: \( 3^2 \) (from 9)
- For 2: \( 2^1 \) (from 6)

Thus,
\[ \text{LCM of denominators} = 3^2 \times 2^1 = 9 \times 2 = 18 \]

### Step 6: Calculate the HCF of the Fractions
The HCF of the fractions is given by:
\[ \text{HCF of fractions} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}} = \frac{1}{18} \]

### Step 7: Calculate the LCM of the Fractions
The LCM of the fractions is given by:
\[ \text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}} = \frac{20}{3} \]

### Step 8: Find the Sum of HCF and LCM
Now we need to find the sum of the HCF and LCM:
\[ \text{Sum} = \frac{1}{18} + \frac{20}{3} \]

To add these fractions, we need a common denominator. The common denominator is 18:
- Convert \( \frac{20}{3} \) to have a denominator of 18:
\[ \frac{20}{3} = \frac{20 \times 6}{3 \times 6} = \frac{120}{18} \]

Now we can add:
\[ \text{Sum} = \frac{1}{18} + \frac{120}{18} = \frac{121}{18} \]

### Final Answer
Thus, the sum of the HCF and LCM of the given fractions is:
\[ \frac{121}{18} \]