If $a : b = c : d$, then $\frac{ma + nc}{mb + nd}$ is equal to

### Concept & The Addendo Property of Proportions If two ratios are equal ($\frac{a}{b} = \frac{c}{d}$), then any linear combination of their numerators divided by the identical linear combination of their denominators will yield the exact same ratio. ### Step-by-Step Solution 1. **Given:** $\frac{a}{b} = \frac{c}{d} = k$ (where $k$ is a constant multiplier). 2. Express the numerators in terms of denominators and the constant: $a = bk$ $c = dk$ 3. Substitute these into the target expression $\frac{ma + nc}{mb + nd}$: $$\frac{m(bk) + n(dk)}{mb + nd}$$ 4. Factor out the common constant $k$ in the numerator: $$\frac{k(mb + nd)}{mb + nd}$$ 5. As long as $mb + nd \neq 0$, the binomials cancel out, leaving: $$k$$ 6. Since we defined $k = \frac{a}{b}$ (which also equals $\frac{c}{d}$), the expression simplifies to $a : b$. ### Exam Strategy & Shortcut Recognize standard mathematical theorems. The theorem states that if $\frac{a}{b} = \frac{c}{d}$, then $\frac{pa + qc}{pb + qd} = \frac{a}{b}$ for any constants $p$ and $q$. Thus, the answer is immediately $a : b$ without any calculation required. ### Common Pitfall Do not assume that the answer involves $m$ and $n$. Because the coefficients $m$ and $n$ are applied symmetrically to both the numerator and the denominator, they act as weights that ultimately factor out of the final proportion. ### Final Answer Therefore, the correct answer is **$a : b$**.