If a : b = c : d = e : f = 1 : 2, then `(pa + qc + re) : (pb + qd + rf)` is equal to :

Options

A. `p : (q + r) `
B. `(p + q) : r`
C. `2 : 3`
D. `1 : 2`

Answer

To solve the problem, we start with the given ratios and express the variables in terms of a common variable.

1. **Given Ratios**: We have \( a : b = c : d = e : f = 1 : 2 \). This means:
- \( \frac{a}{b} = \frac{1}{2} \)
- \( \frac{c}{d} = \frac{1}{2} \)
- \( \frac{e}{f} = \frac{1}{2} \)

2. **Expressing Variables**: From the ratios, we can express \( a, c, e \) in terms of \( b, d, f \):
- \( a = \frac{1}{2}b \)
- \( c = \frac{1}{2}d \)
- \( e = \frac{1}{2}f \)

3. **Substituting into the Expression**: We need to find the ratio \( (pa + qc + re) : (pb + qd + rf) \).
- Substitute \( a, c, e \):
\[ pa + qc + re = p\left(\frac{1}{2}b\right) + q\left(\frac{1}{2}d\right) + r\left(\frac{1}{2}f\right) \]
- This simplifies to:
\[ = \frac{1}{2}(pb + qd + rf) \]

4. **Final Ratio Calculation**: Now, we can express the entire ratio:
\[ \frac{(pa + qc + re)}{(pb + qd + rf)} = \frac{\frac{1}{2}(pb + qd + rf)}{(pb + qd + rf)} \]
- This simplifies to:
\[ = \frac{1}{2} \]

5. **Conclusion**: Therefore, the ratio \( (pa + qc + re) : (pb + qd + rf) \) is equal to \( \frac{1}{2} \).

### Final Answer:
The answer is \( \frac{1}{2} \).