Write an improper fraction using the HCF of 27, 36, 81 as the numerator.

Answer

To solve the question of writing an improper fraction using the HCF of 27, 36, and 81 as the numerator, we will follow these steps:

### Step 1: Find the HCF of 27, 36, and 81
To find the HCF (Highest Common Factor), we will first factorize each number into its prime factors.

- **Factorization of 27**:
- 27 = 3 × 3 × 3 = \(3^3\)

- **Factorization of 36**:
- 36 = 2 × 2 × 3 × 3 = \(2^2 \times 3^2\)

- **Factorization of 81**:
- 81 = 3 × 3 × 3 × 3 = \(3^4\)

Now, we will identify the common prime factors and take the lowest power of each common factor.

- The common prime factor is 3.
- The lowest power of 3 in all three factorizations is \(3^2\).

Thus, the HCF of 27, 36, and 81 is:
\[ HCF = 3^2 = 9 \]

### Step 2: Write an improper fraction using the HCF as the numerator
An improper fraction is defined as a fraction where the numerator is greater than or equal to the denominator. We can use the HCF (9) as the numerator and choose any of the three numbers (27, 36, or 81) as the denominator.

1. **Using 27 as the denominator**:
\[ \text{Improper fraction} = \frac{9}{27} \]

2. **Using 36 as the denominator**:
\[ \text{Improper fraction} = \frac{9}{36} \]

3. **Using 81 as the denominator**:
\[ \text{Improper fraction} = \frac{9}{81} \]

### Step 3: Simplify the fractions
Now, we will simplify each of the improper fractions:

1. **Simplifying \(\frac{9}{27}\)**:
\[ \frac{9 \div 9}{27 \div 9} = \frac{1}{3} \]

2. **Simplifying \(\frac{9}{36}\)**:
\[ \frac{9 \div 9}{36 \div 9} = \frac{1}{4} \]

3. **Simplifying \(\frac{9}{81}\)**:
\[ \frac{9 \div 9}{81 \div 9} = \frac{1}{9} \]

### Final Answer
The improper fractions using the HCF of 27, 36, and 81 as the numerator are:
- \(\frac{9}{27} = \frac{1}{3}\)
- \(\frac{9}{36} = \frac{1}{4}\)
- \(\frac{9}{81} = \frac{1}{9}\)