If $a : b = 5 : 7$ and $c : d = 2a : 3b$, then $ac : bd$ is

### Concept & Substitution in Ratios When asked to find the product of two ratios ($ac : bd$), you can treat them as fractions ($\frac{ac}{bd} = \frac{a}{b} \cdot \frac{c}{d}$) and substitute the known relationships directly. ### Step-by-Step Solution 1. **Given:** $\frac{a}{b} = \frac{5}{7}$ and $\frac{c}{d} = \frac{2a}{3b}$. 2. First, find the exact numerical value of $\frac{c}{d}$ by substituting the value of $\frac{a}{b}$: $$\frac{c}{d} = \frac{2}{3} \cdot \left(\frac{a}{b}\right)$$ $$\frac{c}{d} = \frac{2}{3} \cdot \frac{5}{7} = \frac{10}{21}$$ 3. The question asks for $ac : bd$, which can be written as $\frac{a \cdot c}{b \cdot d}$ or $\frac{a}{b} \cdot \frac{c}{d}$. 4. Multiply the two fractional values: $$\frac{a}{b} \cdot \frac{c}{d} = \frac{5}{7} \cdot \frac{10}{21}$$ $$\frac{ac}{bd} = \frac{50}{147}$$ ### Exam Strategy & Shortcut Recognize that $\frac{ac}{bd}$ is simply the product of the two fractions. Since the second fraction is given in terms of the first, the entire expression condenses to $\frac{a}{b} \cdot \left(\frac{2}{3} \cdot \frac{a}{b}\right) = \frac{2}{3} \cdot \left(\frac{a}{b}\right)^2$. Calculating $\frac{2}{3} \cdot \frac{25}{49}$ instantly gives $\frac{50}{147}$. ### Common Pitfall Misinterpreting $c : d = 2a : 3b$ as fixing absolute values for $a$ and $b$ rather than maintaining the ratio structure, leading to calculation confusion. Stick to fractional multiplication. ### Final Answer Therefore, the correct answer is **$50 : 147$**.