If $a + 2b = 6$ and $ab = 4$, then what is $\frac{2}{a} + \frac{1}{b}$?

### Concept & Formula The core concept here is the algebraic manipulation of fractions to create a form that matches our given equations. By taking the lowest common multiple (LCM) of the denominators, we can merge the terms to reveal a simpler expression. $$ \frac{x}{y} + \frac{u}{v} = \frac{xv + uy}{yv} $$ ### Step-by-Step Solution * **Given:** * Equation 1: $a + 2b = 6$ * Equation 2: $ab = 4$ * Expression to evaluate: $\frac{2}{a} + \frac{1}{b}$ * **Calculation:** 1. Find a common denominator for the target expression $\frac{2}{a} + \frac{1}{b}$. The common denominator is $ab$. 2. Rewrite the fractions with this common denominator: $\frac{2b}{ab} + \frac{a}{ab}$. 3. Combine them into a single fraction: $\frac{a + 2b}{ab}$. 4. Notice that the numerator perfectly matches Equation 1 and the denominator matches Equation 2. 5. Substitute the given values into this new fraction. 6. The numerator $(a + 2b)$ becomes $6$. 7. The denominator $(ab)$ becomes $4$. 8. The resulting fraction is $\frac{6}{4}$. 9. Simplify the fraction by dividing the numerator and denominator by $2$ to get $\frac{3}{2}$. ### Exam Strategy & Shortcut For questions asking for the sum of reciprocals or similar fractional expressions, ALWAYS try simplifying the target expression first before attempting to solve for individual variable values. Solving for $a$ and $b$ using quadratic equations here would be a massive waste of time. Merging the target fractions immediately reveals the answer. ### Common Pitfall A frequent mistake is attempting to isolate $a$ or $b$ from the first equation (e.g., $a = 6 - 2b$) and substituting it into the second to find individual values. This leads to a complex quadratic equation, drastically increasing the risk of calculation errors and wasting valuable exam time. ### Final Answer Therefore, the correct answer is $\frac{3}{2}$.