H.C.F of `1/2, 3/4, 5/6, 7/8,9/10` is -

Options

A. `1/2`
B. `1/(10)`
C. `9/(120)`
D. `1/(120)`

Answer

To find the H.C.F (Highest Common Factor) of the fractions \( \frac{1}{2}, \frac{3}{4}, \frac{5}{6}, \frac{7}{8}, \frac{9}{10} \), we can follow these steps:

### Step 1: Identify the Numerators and Denominators
The numerators of the fractions are \( 1, 3, 5, 7, 9 \) and the denominators are \( 2, 4, 6, 8, 10 \).

### Step 2: Find the H.C.F of the Numerators
Now, we need to find the H.C.F of the numerators \( 1, 3, 5, 7, 9 \).
- The numbers \( 1, 3, 5, 7, 9 \) are all odd numbers, and the only common factor among them is \( 1 \).

### Step 3: Find the H.C.F of the Denominators
Next, we find the H.C.F of the denominators \( 2, 4, 6, 8, 10 \).
- The factors of each denominator are:
- \( 2: 1, 2 \)
- \( 4: 1, 2, 4 \)
- \( 6: 1, 2, 3, 6 \)
- \( 8: 1, 2, 4, 8 \)
- \( 10: 1, 2, 5, 10 \)
- The common factors are \( 1, 2 \), so the H.C.F is \( 2 \).

### Step 4: Calculate the H.C.F of the Fractions
The H.C.F of the fractions can be calculated using the formula:
\[ \text{H.C.F of fractions} = \frac{\text{H.C.F of numerators}}{\text{L.C.M of denominators}} \]
- We have H.C.F of numerators as \( 1 \) and H.C.F of denominators as \( 2 \).
- Therefore, the H.C.F of the fractions is:
\[ \text{H.C.F} = \frac{1}{2} \]

### Final Answer
The H.C.F of \( \frac{1}{2}, \frac{3}{4}, \frac{5}{6}, \frac{7}{8}, \frac{9}{10} \) is \( \frac{1}{2} \).

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