Find the HCF of `( 5)/( 6), ( 15)/( 8) ` and `( 25)/( 32)`

Options

A. 0.052
B. 0.698
C. 0.75
D. Cannot be determined

Answer

To find the HCF (Highest Common Factor) of the fractions \( \frac{5}{6}, \frac{15}{8}, \) and \( \frac{25}{32} \), we can follow these steps:

### Step 1: Identify the Numerators and Denominators
The numerators are 5, 15, and 25, and the denominators are 6, 8, and 32.

### Step 2: Find the HCF of the Numerators
To find the HCF of the numerators (5, 15, and 25):

- The factors of 5 are: 1, 5
- The factors of 15 are: 1, 3, 5, 15
- The factors of 25 are: 1, 5, 25

The common factor among these numbers is 5. Therefore, the HCF of the numerators is:
\[ \text{HCF of } (5, 15, 25) = 5 \]

### Step 3: Find the LCM of the Denominators
Next, we find the LCM of the denominators (6, 8, and 32):

- The prime factorization of 6 is \( 2 \times 3 \)
- The prime factorization of 8 is \( 2^3 \)
- The prime factorization of 32 is \( 2^5 \)

To find the LCM, we take the highest power of each prime factor:
- For 2, the highest power is \( 2^5 \)
- For 3, the highest power is \( 3^1 \)

So, the LCM is:
\[ \text{LCM} = 2^5 \times 3^1 = 32 \times 3 = 96 \]

### Step 4: Combine the HCF and LCM
Now, we can express the HCF of the fractions:
\[ \text{HCF} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}} = \frac{5}{96} \]

### Final Answer
Thus, the HCF of the fractions \( \frac{5}{6}, \frac{15}{8}, \) and \( \frac{25}{32} \) is:
\[ \frac{5}{96} \]