In some quantity of ghee, $60\%$ is pure ghee and $40\%$ is vanaspati. If $10$ kg of pure ghee is added, then the strength of vanaspati ghee becomes $20\%$. The original quantity was

### Concept & Logic When only pure ghee is added to the mixture, the absolute quantity of the other ingredient (vanaspati) remains strictly constant. Setting up an equation based on this **Constant Component** is the fastest path to the solution. $$ \text{Initial Mass of Vanaspati} = \text{Final Mass of Vanaspati} $$ ### Step-by-Step Solution * **Given:** * Initial pure ghee = $60\%$, meaning initial vanaspati = $40\%$ * Added pure ghee = $10$ kg * Final strength of vanaspati = $20\%$ * **Calculation:** * Let the original quantity of the mixture be $x$ kg. * The absolute mass of vanaspati in the original mixture is $40\%$ of $x = 0.40x$. * After adding $10$ kg of pure ghee, the new total quantity of the mixture becomes $(x + 10)$ kg. * In this new mixture, the vanaspati constitutes $20\%$. So, its mass is $20\%$ of $(x + 10) = 0.20(x + 10)$. * Equating the constant vanaspati mass: $0.40x = 0.20(x + 10)$. * Divide both sides by $0.20$: $2x = x + 10$. * Subtract $x$ from both sides: $x = 10$ kg. ### Exam Strategy & Shortcut **Inverse Ratio Shortcut:** Focus entirely on the constant component (vanaspati). Its percentage drops from $40\%$ to $20\%$, which is a ratio of $2 : 1$. Because the absolute mass of vanaspati is constant, the total mixture weight must grow in the exact inverse ratio, which is $1 : 2$. If the original weight is $1$ unit and the new weight is $2$ units, the difference ($1$ unit) represents the added ghee. We are given that $1$ unit (added ghee) $= 10$ kg. Therefore, the original quantity ($1$ unit) must also be $10$ kg. ### Common Pitfall Students often waste time tracking the pure ghee: trying to solve $0.60x + 10 = 0.80(x + 10)$. While mathematically valid, it involves more complex arithmetic and terms on both sides of the equation, greatly increasing the chance of an algebra error compared to using the constant vanaspati. ### Final Answer **Therefore, the correct answer is 10 kg.**