40% of 60% of 32% of an amount is ₹ 432. What is the amount?

### Concept & Formula This requires solving a successive percentage chain. Converting percentages to simplified fractions before solving the algebraic equation will drastically reduce the risk of decimal multiplication errors. $$ \text{Final Amount} = \text{Base} \times \left( \frac{P_1}{100} \right) \times \left( \frac{P_2}{100} \right) \times \left( \frac{P_3}{100} \right) $$ ### Step-by-Step Solution * **Given:** Equation: $ 40\% \text{ of } 60\% \text{ of } 32\% \text{ of Amount } (A) = 432 $ * **Calculation:** Step 1: Convert all percentages into their simplest fractional forms. $ 40\% = \frac{40}{100} = \frac{2}{5} $ $ 60\% = \frac{60}{100} = \frac{3}{5} $ $ 32\% = \frac{32}{100} = \frac{8}{25} $ Step 2: Set up the equation with the fractions. $ \left( \frac{2}{5} \right) \times \left( \frac{3}{5} \right) \times \left( \frac{8}{25} \right) \times A = 432 $ Step 3: Multiply the numerators and denominators together. $ \left( \frac{2 \times 3 \times 8}{5 \times 5 \times 25} \right) \times A = 432 $ $ \left( \frac{48}{625} \right) \times A = 432 $ Step 4: Solve for $A$ by isolating it. $ A = 432 \times \left( \frac{625}{48} \right) $ $ A = 9 \times 625 $ $ A = 5625 $ Since ₹ 5625 is not explicitly listed among options (a), (b), or (c), the correct choice must be "None of these". ### Exam Strategy & Shortcut Look for clean multiples. Notice that the numerator of the combined fraction is 48 ($ 4 \times 6 \times 2 $ dropping zeroes). You have to divide 432 by 48. $ 432 \div 48 $. Notice that $ 48 \times 10 = 480 $. 432 is exactly 48 less than 480. So, $ 432 \div 48 $ is exactly 9. Then you just multiply 9 by the combined denominator ($ 5 \times 5 \times 25 = 625 $). $ 9 \times 625 = 5625 $. This mental math shortcut avoids writing down a massive decimal chain like $ 0.4 \times 0.6 \times 0.32 = 0.0768 $. ### Common Pitfall The most common trap is attempting to multiply the decimals directly ($ 0.4 \times 0.6 \times 0.32 = 0.0768 $) and then attempting long division ($ 432 \div 0.0768 $) by hand. This wastes tremendous time and frequently leads to misplaced decimal points in the final answer. ### Final Answer **Therefore, the correct answer is None of these.**