Express `frac(-12)(13)` as a rational number with numerator
(i) -48,
(ii) 60.

Answer

To express \(-\frac{12}{13}\) as a rational number with specific numerators, we will follow the steps below for each case.

### Part (i): Express \(-\frac{12}{13}\) with numerator \(-48\)

1. **Identify the relationship**: We want to express \(-\frac{12}{13}\) in the form \(-\frac{48}{q}\), where \(q\) is the new denominator we need to find.

2. **Find the multiplier**: To change the numerator from \(-12\) to \(-48\), we need to determine what we multiply \(-12\) by:
\[ -12 \times k = -48 \]
Solving for \(k\):
\[ k = \frac{-48}{-12} = 4 \]

3. **Multiply both numerator and denominator by the same number**: We multiply both the numerator and the denominator of \(-\frac{12}{13}\) by \(4\):
\[ -\frac{12 \times 4}{13 \times 4} = -\frac{48}{52} \]

4. **Conclusion**: Thus, \(-\frac{12}{13}\) can be expressed as:
\[ -\frac{48}{52} \]

### Part (ii): Express \(-\frac{12}{13}\) with numerator \(60\)

1. **Identify the relationship**: We want to express \(-\frac{12}{13}\) in the form \(-\frac{60}{q}\).

2. **Find the multiplier**: To change the numerator from \(-12\) to \(60\), we need to determine what we multiply \(-12\) by:
\[ -12 \times k = 60 \]
Solving for \(k\):
\[ k = \frac{60}{-12} = -5 \]

3. **Multiply both numerator and denominator by the same number**: We multiply both the numerator and the denominator of \(-\frac{12}{13}\) by \(-5\):
\[ -\frac{12 \times -5}{13 \times -5} = -\frac{60}{-65} \]

4. **Conclusion**: Thus, \(-\frac{12}{13}\) can be expressed as:
\[ -\frac{60}{-65} \]

### Summary of Results:
- For numerator \(-48\): \(-\frac{12}{13} = -\frac{48}{52}\)
- For numerator \(60\): \(-\frac{12}{13} = -\frac{60}{-65}\)