Express `-48/36` as a rational number with numerator (i) -4(ii) 8

Answer

To express \(-\frac{48}{36}\) as a rational number with different specified numerators, we will follow these steps:

### Part (i): Expressing with numerator -4

1. **Identify the original fraction**: We start with the fraction \(-\frac{48}{36}\).

2. **Determine the factor to multiply**: To change the numerator from \(-48\) to \(-4\), we need to find a factor. We can do this by dividing \(-48\) by \(-4\):
\[ \text{Factor} = \frac{-4}{-48} = \frac{1}{12} \]
Thus, we will multiply the original fraction by \(\frac{1}{12}\).

3. **Multiply the fraction**: We multiply both the numerator and the denominator by \(12\):
\[ -\frac{48}{36} \times \frac{1}{1} = -\frac{48 \times 1}{36 \times 1} = -\frac{48}{36} \]
However, to achieve the desired numerator of \(-4\), we will multiply by \(\frac{12}{12}\):
\[ -\frac{48}{36} \times \frac{1}{12} = -\frac{48 \div 12}{36 \div 12} = -\frac{4}{3} \]

4. **Final answer for part (i)**: Thus, the rational number with numerator \(-4\) is:
\[ -\frac{4}{3} \]

### Part (ii): Expressing with numerator 8

1. **Identify the original fraction**: We start again with the fraction \(-\frac{48}{36}\).

2. **Determine the factor to multiply**: To change the numerator from \(-48\) to \(8\), we need to find a factor. We can do this by dividing \(8\) by \(-48\):
\[ \text{Factor} = \frac{8}{-48} = -\frac{1}{6} \]
Thus, we will multiply the original fraction by \(-\frac{1}{6}\).

3. **Multiply the fraction**: We multiply both the numerator and the denominator by \(-6\):
\[ -\frac{48}{36} \times \frac{-1}{-1} = -\frac{48 \times -1}{36 \times -1} = \frac{48}{36} \]
To achieve the desired numerator of \(8\), we will multiply by \(-\frac{1}{6}\):
\[ -\frac{48}{36} \times \frac{-1}{-1} = \frac{8}{-6} \]

4. **Final answer for part (ii)**: Thus, the rational number with numerator \(8\) is:
\[ \frac{8}{-6} \text{ or } -\frac{8}{6} \text{ which simplifies to } -\frac{4}{3} \]

### Summary of Answers:
- For part (i): \(-\frac{4}{3}\)
- For part (ii): \(\frac{8}{-6}\) or \(-\frac{4}{3}\)