Which of the following pairs represent equivalent rational numbers ?

Options

A. `(1)/(4) and (-4)/(-16)`
B. `(-2)/(3) and (8)/(12)`
C. `(12)/(15) and (10)/(18)`
D. `(27)/(54) and (3)/(2)`

Answer

To determine which pairs represent equivalent rational numbers, we will analyze each option by converting the fractions to their lowest forms and checking for equality.

### Step-by-Step Solution:

1. **Identify the pairs of rational numbers**:
- Option A: \( \frac{1}{4} \) and \( -\frac{4}{-16} \)
- Option B: \( -\frac{2}{3} \) and \( \frac{8}{12} \)
- Option C: \( \frac{12}{15} \) and \( \frac{10}{18} \)
- Option D: \( \frac{27}{54} \) and \( \frac{3}{2} \)

2. **Analyze Option A**:
- First number: \( \frac{1}{4} \) is already in its lowest form.
- Second number: \( -\frac{4}{-16} \)
- The negatives cancel out: \( -\frac{4}{-16} = \frac{4}{16} \)
- Simplifying \( \frac{4}{16} \): Divide both numerator and denominator by 4:
\[ \frac{4 \div 4}{16 \div 4} = \frac{1}{4} \]
- Since both are \( \frac{1}{4} \), they are equivalent.

3. **Analyze Option B**:
- First number: \( -\frac{2}{3} \) is already in its lowest form.
- Second number: \( \frac{8}{12} \)
- Simplifying \( \frac{8}{12} \): Divide both by 4:
\[ \frac{8 \div 4}{12 \div 4} = \frac{2}{3} \]
- Now we compare \( -\frac{2}{3} \) and \( \frac{2}{3} \). They are not equal due to the negative sign, so they are not equivalent.

4. **Analyze Option C**:
- First number: \( \frac{12}{15} \)
- Simplifying \( \frac{12}{15} \): Divide both by 3:
\[ \frac{12 \div 3}{15 \div 3} = \frac{4}{5} \]
- Second number: \( \frac{10}{18} \)
- Simplifying \( \frac{10}{18} \): Divide both by 2:
\[ \frac{10 \div 2}{18 \div 2} = \frac{5}{9} \]
- Now we compare \( \frac{4}{5} \) and \( \frac{5}{9} \). They are not equal, so they are not equivalent.

5. **Analyze Option D**:
- First number: \( \frac{27}{54} \)
- Simplifying \( \frac{27}{54} \): Divide both by 27:
\[ \frac{27 \div 27}{54 \div 27} = \frac{1}{2} \]
- Second number: \( \frac{3}{2} \) is already in its lowest form.
- Now we compare \( \frac{1}{2} \) and \( \frac{3}{2} \). They are not equal, so they are not equivalent.

### Conclusion:
The only equivalent pair of rational numbers is **Option A: \( \frac{1}{4} \) and \( -\frac{4}{-16} \)**.