40% of 4.5 + $x$% of $\frac{2}{3}$ = 20% of 10

### Concept & Formula This problem requires evaluating a linear equation containing percentages and fractions. We must simplify the known terms on both sides of the equation to isolate the unknown percentage variable. $$ \text{Value}_1 + \left( \frac{x}{100} \right) \times \text{Base} = \text{Value}_2 $$ ### Step-by-Step Solution * **Given:** Equation: 40% of 4.5 + $x$% of $ \frac{2}{3} $ = 20% of 10 * **Calculation:** Step 1: Simplify the first term (40% of 4.5). $ 40\% = \frac{4}{10} = 0.4 $ $ 0.4 \times 4.5 = 1.8 $ Step 2: Simplify the term on the right side of the equation (20% of 10). $ 20\% \text{ of } 10 = 2 $ Step 3: Substitute the simplified values back into the equation. $ 1.8 + \left( \frac{x}{100} \right) \times \left( \frac{2}{3} \right) = 2 $ Step 4: Isolate the term containing $x$. $ \left( \frac{2x}{300} \right) = 2 - 1.8 $ $ \left( \frac{2x}{300} \right) = 0.2 $ Step 5: Solve for $x$. $ 2x = 0.2 \times 300 $ $ 2x = 60 $ $ x = 30 $ ### Exam Strategy & Shortcut Use mental math to speed up the process. For 40% of 4.5: double 4.5 to get 9 (which is 20%), then double again to get 1.8 (40%). For 20% of 10: easily 2. The difference is $ 2 - 1.8 = 0.2 $. So, what percentage of $ \frac{2}{3} $ gives 0.2? Convert 0.2 to a fraction: $ \frac{1}{5} $. $ x\% \times \left(\frac{2}{3}\right) = \frac{1}{5} \implies x\% = \left(\frac{1}{5}\right) \times \left(\frac{3}{2}\right) = \frac{3}{10} = 30\% $. ### Common Pitfall Students often struggle with mixing decimals (4.5) and fractions ($ \frac{2}{3} $). A common error is converting $ \frac{2}{3} $ to 0.666..., which introduces rounding errors and makes isolating $x$ extremely difficult. Always keep fractions as fractions until the final step. ### Final Answer **Therefore, the correct answer is 30.**