Discussion
A dishonest milkman professes to sell his milk at cost price but mixes it with water and gains 25%. What is the ratio of water to milk in the mixture?
Answer: A
Correct answer: 1 : 4
The gain arises entirely from the water, which costs nothing but is sold at the price of milk. Let the cost of one litre of milk be 1. If the milkman sells \( 1 + x \) litres of mixture containing 1 litre of milk and \( x \) of water, his cost is 1 and his receipt is \( 1 + x \), so \[ \text{gain} = \frac{x}{1} = 25\% = \frac{1}{4} \;\Rightarrow\; x = \frac{1}{4}. \] The ratio of water to milk is therefore \( \frac{1}{4} : 1 = \mathbf{1 : 4} \). The useful shortcut is that the ratio of water to milk equals the gain expressed as a fraction: a 25% gain means 1 : 4, a 20% gain means 1 : 5, and a 50% gain means 1 : 2. The step that trips people is reading the ratio the right way round - the question asks for water to milk, and reversing it gives 4 : 1, which is also offered in many versions of this question.
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