In a certain quantity of ghee, 60% is pure ghee and 40% is vanaspati. If 10 kilograms of pure ghee is added to this mixture, the proportion of vanaspati becomes 20% of the new mixture. What was the original quantity of the mixture?

Introduction:
This question belongs to the topic of mixtures and alligation involving percentages. A mixture of pure ghee and vanaspati is strengthened by adding more pure ghee, leading to a change in the percentage of vanaspati. The task is to back calculate the original quantity of the mixture from the new composition.

Given Data / Assumptions:
Original mixture has 60% pure ghee and 40% vanaspati.
Let original quantity of mixture be W kilograms.
Pure ghee initially = 60% of W, vanaspati initially = 40% of W.
10 kilograms of pure ghee are added to the mixture.
After addition, vanaspati becomes 20% of the new mixture.
We must find the original quantity W in kilograms.

Concept / Approach:
Because only pure ghee is added, the amount of vanaspati remains unchanged, while the total quantity and the amount of pure ghee both increase. The vanaspati percentage in the new mixture is its unchanged quantity divided by the new total quantity. Setting this equal to 20% and solving for W gives the original quantity. This is a straightforward application of the percentage formula for mixtures.

Step-by-Step Solution:
Let original quantity of mixture be W kilograms. Original pure ghee = 60% of W = 0.60 * W. Original vanaspati = 40% of W = 0.40 * W. We add 10 kilograms of pure ghee to the mixture. New amount of pure ghee = 0.60 * W + 10. Amount of vanaspati remains 0.40 * W because we did not add or remove any vanaspati. New total quantity = W + 10 kilograms. It is given that the strength of vanaspati becomes 20% of the new mixture. So vanaspati percentage = (vanaspati quantity / total quantity) * 100 = 20%. Write this as an equation: (0.40 * W) / (W + 10) * 100 = 20. Simplify by dividing both sides by 20: (0.40 * W) / (W + 10) = 0.20. Multiply both sides by (W + 10): 0.40 * W = 0.20 * (W + 10). Expand the right side: 0.40 * W = 0.20 * W + 2. Subtract 0.20 * W from both sides: 0.20 * W = 2. So W = 2 / 0.20 = 10 kilograms.
Verification / Alternative check:
Verify using W = 10 kilograms. Initially pure ghee = 60% of 10 = 6 kilograms and vanaspati = 40% of 10 = 4 kilograms. After adding 10 kilograms of pure ghee, new pure ghee amount = 6 + 10 = 16 kilograms. Total mixture = 10 + 10 = 20 kilograms. Vanaspati still = 4 kilograms. Vanaspati percentage in the new mixture = (4 / 20) * 100 = 20%, which matches the condition in the question. This confirms that the original quantity was 10 kilograms.

Why Other Options Are Wrong:
If the original quantity were 15 kilograms, 20 kilograms, 25 kilograms, or 30 kilograms, repeating the same calculations would not yield a final vanaspati percentage of exactly 20%. Each of those values would lead to a different final percentage. Therefore they do not satisfy the condition given in the problem.

Common Pitfalls:
A common mistake is to assume that the percentage of pure ghee changes linearly with the amount added, without considering the change in total quantity. Another pitfall is to misinterpret the statement and treat 20% as the new pure ghee percentage instead of the vanaspati percentage. Carefully tracking which component s percentage is being discussed and how total quantity changes is essential in mixture problems.

Final Answer:
The original quantity of the mixture was 10 kg.