Express `-6/11` as a rational number with numerator (i) -36 (ii) 42

Answer

To express \(-\frac{6}{11}\) as a rational number with different numerators, we will follow these steps:

### Part (i): Express \(-\frac{6}{11}\) with numerator \(-36\)

1. **Identify the desired numerator**: We want to change the numerator from \(-6\) to \(-36\).

2. **Determine the multiplication factor**: To find out what we need to multiply \(-6\) by to get \(-36\), we can set up the equation:
\[ -6 \times x = -36 \]
Solving for \(x\):
\[ x = \frac{-36}{-6} = 6 \]

3. **Multiply both the numerator and denominator by this factor**: We multiply \(-\frac{6}{11}\) by \(\frac{6}{6}\):
\[ -\frac{6}{11} \times \frac{6}{6} = \frac{-6 \times 6}{11 \times 6} = \frac{-36}{66} \]

### Part (ii): Express \(-\frac{6}{11}\) with numerator \(42\)

1. **Identify the desired numerator**: Now we want to change the numerator from \(-6\) to \(42\).

2. **Determine the multiplication factor**: To find out what we need to multiply \(-6\) by to get \(42\), we can set up the equation:
\[ -6 \times y = 42 \]
Solving for \(y\):
\[ y = \frac{42}{-6} = -7 \]

3. **Multiply both the numerator and denominator by this factor**: We multiply \(-\frac{6}{11}\) by \(\frac{-7}{-7}\):
\[ -\frac{6}{11} \times \frac{-7}{-7} = \frac{-6 \times -7}{11 \times -7} = \frac{42}{-77} \]

### Final Answers:
- For numerator \(-36\): \(-\frac{36}{66}\)
- For numerator \(42\): \(\frac{42}{-77}\)