If $\frac{m}{n} = \frac{4}{3}$ and $\frac{r}{t} = \frac{9}{14}$, the value of $\frac{3mr - nt}{4nt - 7mr}$ is

### Concept & Formula This problem evaluates homogeneous algebraic expressions. Because each term in both the numerator and denominator consists of two variables multiplied together (degree 2), we can convert the entire expression into a function of the combined ratio $\frac{mr}{nt}$. $$ \text{Factoring ratios: } \frac{a \cdot c}{b \cdot d} = \left(\frac{a}{b}\right) \times \left(\frac{c}{d}\right) $$ ### Step-by-Step Solution * **Given:** * Ratio 1: $\frac{m}{n} = \frac{4}{3}$ * Ratio 2: $\frac{r}{t} = \frac{9}{14}$ * Target expression: $\frac{3mr - nt}{4nt - 7mr}$ * **Calculation:** 1. First, find the value of the combined product ratio $\frac{mr}{nt}$. $\frac{mr}{nt} = \left(\frac{m}{n}\right) \times \left(\frac{r}{t}\right) = \left(\frac{4}{3}\right) \times \left(\frac{9}{14}\right)$. 2. Simplify the multiplication: $\frac{4 \times 9}{3 \times 14} = \frac{36}{42} = \frac{6}{7}$. 3. Now, manipulate the target expression so that it relies purely on this known ratio $\frac{mr}{nt}$. Divide every term in the numerator and the denominator by $nt$. $\frac{ \frac{3mr}{nt} - \frac{nt}{nt} }{ \frac{4nt}{nt} - \frac{7mr}{nt} }$ 4. Simplify the expression: $\frac{ 3\left(\frac{mr}{nt}\right) - 1 }{ 4 - 7\left(\frac{mr}{nt}\right) }$ 5. Substitute the calculated value $\frac{mr}{nt} = \frac{6}{7}$ into the simplified expression. $\frac{ 3\left(\frac{6}{7}\right) - 1 }{ 4 - 7\left(\frac{6}{7}\right) }$ 6. Simplify the numerator: $\frac{18}{7} - 1 = \frac{18}{7} - \frac{7}{7} = \frac{11}{7}$. 7. Simplify the denominator: $4 - \frac{42}{7} = 4 - 6 = -2$. 8. Divide the numerator by the denominator: $\frac{ \frac{11}{7} }{ -2 } = -\frac{11}{14}$. ### Exam Strategy & Shortcut Instead of complex algebraic factoring, use the Direct Value Substitution method since the expression is homogeneous. Assume $m = 4$, $n = 3$ and $r = 9$, $t = 14$. Plug these directly into $\frac{3mr - nt}{4nt - 7mr}$: Numerator: $3(4)(9) - (3)(14) = 108 - 42 = 66$. Denominator: $4(3)(14) - 7(4)(9) = 168 - 252 = -84$. Result: $\frac{66}{-84}$. Divide top and bottom by $6$: $\frac{11}{-14} = -\frac{11}{14}$. This is faster and avoids fraction arithmetic. ### Common Pitfall A standard pitfall is failing to recognize that you can divide by $nt$ to create the $\frac{mr}{nt}$ ratio, leading students to attempt to solve for individual variables ($m, n, r, t$) which is impossible as there are 4 variables and only 2 equations. ### Final Answer Therefore, the correct answer is $-\frac{11}{14}$.