If $x : y = 3 : 4$ and $a : b = 1 : 2$, then the value of $\frac{2xa + yb}{3yb - 4xa}$ is

### Concept & Independent Ratio Substitution When evaluating an expression that consists of products of variables from two independent, unrelated ratios, you can directly substitute the relative values of each variable as long as the expression is homogeneous with respect to both sets of variables. ### Step-by-Step Solution - **Given:** $x : y = 3 : 4$ and $a : b = 1 : 2$. - **Substitute:** Let $x = 3, y = 4$ and $a = 1, b = 2$. - **Target Expression:** $\frac{2xa + yb}{3yb - 4xa}$ - **Calculate individual terms:** - $xa = 3 \times 1 = 3 \implies 2xa = 2 \times 3 = 6$ - $yb = 4 \times 2 = 8 \implies yb = 8$ - $3yb = 3 \times 8 = 24$ - $4xa = 4 \times 3 = 12$ - **Substitute into fraction:** - Numerator: $6 + 8 = 14$ - Denominator: $24 - 12 = 12$ - **Simplify:** $\frac{14}{12} = \frac{7}{6}$. ### Exam Strategy & Shortcut Since every term in the numerator and denominator is a product of one variable from the first set $(x, y)$ and one from the second set $(a, b)$, the proportionality constants ($k$ and $m$) will appear as $km$ in every term and factor out completely. Plug in the numbers immediately: $(6+8)/(24-12) = 14/12 = 7/6$. ### Common Pitfall Mixing up the variables during substitution (e.g., multiplying $x$ with $b$ instead of $a$) is the most frequent error here due to reading too quickly. ### Final Answer Therefore, the correct answer is **$\frac{7}{6}$**.