If $\frac{a}{b} = \frac{4}{3}$, then $\frac{3a+2b}{3a-2b} = x$

### Concept & Formula This problem tests the concept of ratio substitution in homogeneous equations. Because both the numerator and the denominator are linear combinations of $a$ and $b$ (degree 1), the absolute values of $a$ and $b$ do not matter—only their ratio matters. $$ \text{When } \frac{a}{b} = k, \text{ substitute directly: } a = 4, b = 3 $$ ### Step-by-Step Solution * **Given:** * Ratio: $\frac{a}{b} = \frac{4}{3}$ * Expression to evaluate: $\frac{3a + 2b}{3a - 2b}$ * **Calculation (Method 1: Direct Substitution):** 1. Since the expression is homogeneous, we can assume proportional values directly. Let $a = 4$ and $b = 3$. 2. Substitute $a = 4$ and $b = 3$ into the numerator: $3(4) + 2(3) = 12 + 6 = 18$ 3. Substitute $a = 4$ and $b = 3$ into the denominator: $3(4) - 2(3) = 12 - 6 = 6$ 4. Divide the resulting numerator by the denominator: $\frac{18}{6} = 3$. * **Calculation (Method 2: Algebraic Manipulation):** 1. Divide every term in the numerator and denominator by $b$. This forces the expression into terms of $(\frac{a}{b})$. 2. $\frac{ \frac{3a}{b} + \frac{2b}{b} }{ \frac{3a}{b} - \frac{2b}{b} }$ 3. Simplify the $b$ terms: $\frac{ 3(\frac{a}{b}) + 2 }{ 3(\frac{a}{b}) - 2 }$ 4. Substitute the given value $\frac{a}{b} = \frac{4}{3}$ into the equation. 5. $\frac{ 3(\frac{4}{3}) + 2 }{ 3(\frac{4}{3}) - 2 }$ 6. The $3$s cancel out: $\frac{4 + 2}{4 - 2} = \frac{6}{2} = 3$. ### Exam Strategy & Shortcut Direct Substitution (Method 1) is the fastest and most reliable shortcut for exams. Whenever you are given a ratio $\frac{a}{b} = \frac{x}{y}$ and asked to evaluate a fraction where every term has the same degree of variables (homogeneous), NEVER use a constant $k$ (e.g., $a=4k, b=3k$). Just plug in the numbers directly. ### Common Pitfall A common mistake is incorrectly executing the algebraic division if attempting Method 2, such as dividing the numerator by $a$ and the denominator by $b$, which breaks the fraction. Always ensure whatever operation you perform on the top is identically mirrored on the bottom. ### Final Answer Therefore, the correct answer is 3.