If $a : b = c : d = e : f = 1 : 2$, then $(3a + 5c + 7e) : (3b + 5d + 7f)$ is equal to

### Concept & Proportional Scaling When multiple ratios are equal to a constant, any linear combination of the numerator terms divided by the exact same linear combination of the denominator terms maintains the identical ratio. This is a direct consequence of factoring out the constant multiplier. ### Step-by-Step Solution 1. **Given:** $\frac{a}{b} = \frac{c}{d} = \frac{e}{f} = \frac{1}{2}$ 2. From this, we can express the numerator variables in terms of their respective denominators: $a = \frac{1}{2}b$ $c = \frac{1}{2}d$ $e = \frac{1}{2}f$ 3. We need to evaluate the expression $\frac{3a + 5c + 7e}{3b + 5d + 7f}$. 4. Substitute the values of $a$, $c$, and $e$ into the numerator: $$= \frac{3(\frac{1}{2}b) + 5(\frac{1}{2}d) + 7(\frac{1}{2}f)}{3b + 5d + 7f}$$ 5. Factor out the common multiplier $\frac{1}{2}$ from the numerator terms: $$= \frac{\frac{1}{2}(3b + 5d + 7f)}{3b + 5d + 7f}$$ 6. Cancel out the common binomial factor $(3b + 5d + 7f)$: $$= \frac{1}{2}$$ 7. Expressing this as a ratio gives $1 : 2$. ### Exam Strategy & Shortcut Recognize the structural symmetry. If $\frac{a}{b} = \frac{c}{d} = \frac{e}{f} = k$, then $\frac{ma + nc + pe}{mb + nd + pf} = k$ for any scalar values $m, n, p$. Because the coefficients $3, 5, 7$ perfectly match top and bottom, the ratio remains exactly the same as the original given ratio: $1 : 2$. ### Common Pitfall Students often waste time trying to assign arbitrary individual values to $a, b, c, d, e, f$ (like $a=1, b=2, c=2, d=4...$) and calculating the heavy arithmetic $3(1) + 5(2) + 7(3)...$ which is prone to addition errors. Rely on the algebraic property instead. ### Final Answer Therefore, the correct answer is **$1 : 2$**.