40% of 60% of $\frac{3}{5}$th of a number is 504. What is 25% of $\frac{2}{5}$th of that number?

### Concept & Formula This problem involves establishing an algebraic equation by converting percentages to decimals or fractions. Once the unknown base number is found, a second set of operations is applied to it. $$ \text{Value} = \text{Base Number} \times \text{Fraction}_1 \times \text{Fraction}_2 \times \dots $$ ### Step-by-Step Solution * **Given:** Condition 1: 40% of 60% of $ \frac{3}{5} $ of Number ($x$) = 504 Target: Find 25% of $ \frac{2}{5} $ of Number ($x$) * **Calculation:** Step 1: Set up the equation to find $x$. Convert percentages to fractions for easier cancellation. $ 40\% = \frac{2}{5} $ $ 60\% = \frac{3}{5} $ Equation: $ \frac{2}{5} \times \frac{3}{5} \times \frac{3}{5} \times x = 504 $ Step 2: Solve for $x$. $ \frac{18}{125} \times x = 504 $ $ x = \frac{504 \times 125}{18} $ $ x = 28 \times 125 $ $ x = 3500 $ Step 3: Calculate the target value using the determined $x$. Target = 25% of $ \frac{2}{5} $ of 3500 $ 25\% = \frac{1}{4} $ Target = $ \frac{1}{4} \times \frac{2}{5} \times 3500 $ Target = $ \frac{2}{20} \times 3500 $ Target = $ \frac{1}{10} \times 3500 $ Target = $ 350 $ ### Exam Strategy & Shortcut Instead of calculating the full value of $x$, find the ratio of the target expression to the given expression. Given Expression (G) = $ 0.4 \times 0.6 \times 0.6 \times x = 0.144x $ Target Expression (T) = $ 0.25 \times 0.4 \times x = 0.100x $ Set up a proportion: $ \frac{T}{G} = \frac{0.100}{0.144} $ $ T = 504 \times \left( \frac{100}{144} \right) $ $ T = 504 \times \left( \frac{25}{36} \right) $ $ T = 14 \times 25 = 350 $ This ratio method entirely bypasses the need to find the large intermediate number 3500. ### Common Pitfall Converting percentages to decimals (0.4, 0.6) and mixing them with fractions ($ \frac{3}{5} $) often leads to calculation errors due to messy decimal arithmetic. It is almost always safer and faster to convert all terms to simplified fractions ($ \frac{2}{5}, \frac{3}{5} $) so numerators and denominators can be easily cross-cancelled. ### Final Answer **Therefore, the correct answer is 350.**