Find the equivalent fraction of `(5)/(9)` having
(i) denominater 54
(ii) numerator 35

Answer

To find the equivalent fractions of \( \frac{5}{9} \) with the specified conditions, we will solve it step by step.

### Part (i): Find the equivalent fraction of \( \frac{5}{9} \) with a denominator of 54.

1. **Identify the original fraction**: The original fraction is \( \frac{5}{9} \).
2. **Determine the factor to multiply the denominator**: We need to find out what we can multiply the denominator (9) by to get 54.
\[ 9 \times ? = 54 \]
To find the factor, we divide 54 by 9:
\[ ? = \frac{54}{9} = 6 \]
3. **Multiply both the numerator and denominator by this factor**: Now that we know we need to multiply by 6, we do the following:
\[ \text{New numerator} = 5 \times 6 = 30 \]
\[ \text{New denominator} = 9 \times 6 = 54 \]
4. **Write the equivalent fraction**: Therefore, the equivalent fraction of \( \frac{5}{9} \) with a denominator of 54 is:
\[ \frac{30}{54} \]

### Part (ii): Find the equivalent fraction of \( \frac{5}{9} \) with a numerator of 35.

1. **Identify the original fraction**: The original fraction is \( \frac{5}{9} \).
2. **Determine the factor to multiply the numerator**: We need to find out what we can multiply the numerator (5) by to get 35.
\[ 5 \times ? = 35 \]
To find the factor, we divide 35 by 5:
\[ ? = \frac{35}{5} = 7 \]
3. **Multiply both the numerator and denominator by this factor**: Now that we know we need to multiply by 7, we do the following:
\[ \text{New numerator} = 5 \times 7 = 35 \]
\[ \text{New denominator} = 9 \times 7 = 63 \]
4. **Write the equivalent fraction**: Therefore, the equivalent fraction of \( \frac{5}{9} \) with a numerator of 35 is:
\[ \frac{35}{63} \]

### Summary of Results:
- The equivalent fraction of \( \frac{5}{9} \) with a denominator of 54 is \( \frac{30}{54} \).
- The equivalent fraction of \( \frac{5}{9} \) with a numerator of 35 is \( \frac{35}{63} \).