Express `frac(4)(7)` as a rational number with denominator
(i) 21,
(ii) -35.

Answer

To express the rational number \( \frac{4}{7} \) with different denominators, we will follow these steps:

### Step 1: Express \( \frac{4}{7} \) with a denominator of 21
1. **Identify the target denominator**: We want to convert \( \frac{4}{7} \) to have a denominator of 21.
2. **Find the factor**: Calculate how many times 7 goes into 21:
\[ 21 \div 7 = 3 \]
3. **Multiply both the numerator and the denominator by this factor**:
\[ \frac{4 \times 3}{7 \times 3} = \frac{12}{21} \]

### Step 2: Express \( \frac{4}{7} \) with a denominator of -35
1. **Identify the target denominator**: Now we want to express \( \frac{4}{7} \) with a denominator of -35.
2. **Find the factor**: Calculate how many times 7 goes into 35:
\[ 35 \div 7 = 5 \]
3. **Multiply both the numerator and the denominator by -5** (to ensure the denominator is negative):
\[ \frac{4 \times -5}{7 \times -5} = \frac{-20}{-35} \]

### Final Answers:
- For a denominator of 21: \( \frac{4}{7} = \frac{12}{21} \)
- For a denominator of -35: \( \frac{4}{7} = \frac{-20}{-35} \)

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