Paint is required to be thinned so that the final mixture has paint and water in the ratio 2 : 1.5. Due to a mistake, a painter has 6 litres of mixture that is exactly half paint and half water. What should he add to obtain the correct proportions?

Introduction / Context:
This question is about mixtures and ratios. The painter wants a final mixture with a specified ratio of paint to water, but the current mixture has equal amounts of paint and water. We must determine what to add so that the corrected mixture satisfies the required ratio. Such mixture questions test understanding of ratio adjustment and proportional reasoning.

Given Data / Assumptions:
• Required ratio of paint : water is 2 : 1.5.
• The painter mistakenly has 6 litres of mixture with half paint and half water.
• Therefore, current paint = 3 litres and current water = 3 litres.
• We are allowed to add only paint, only water, or a combination, as in the options.

Concept / Approach:
First, simplify the desired ratio. The ratio 2 : 1.5 can be scaled by 2 to remove the decimal, giving 4 : 3. This means that in the final mixture, paint : water should be 4 : 3. Let the extra paint added be p litres and extra water added be w litres. Then form an equation so that (3 + p) : (3 + w) = 4 : 3 and check which option satisfies this relationship. We will see that only adding 1 litre of paint and no extra water yields the desired ratio.

Step-by-Step Solution:
Step 1: Simplify the required ratio 2 : 1.5 to 4 : 3 by multiplying both terms by 2.Step 2: Currently, the mixture has 3 litres of paint and 3 litres of water.Step 3: Suppose we add only paint as in option A: new paint = 3 + 1 = 4 litres, new water = 3 litres.Step 4: The new ratio becomes paint : water = 4 : 3, which matches the required proportion.Step 5: Check that no other option is needed since we already meet the exact target ratio by this simple adjustment.

Verification / Alternative check:
If we consider the general condition (3 + p) : (3 + w) = 4 : 3, cross-multiplying gives 3(3 + p) = 4(3 + w). That is 9 + 3p = 12 + 4w. For option A, p = 1 and w = 0, so left side is 9 + 3 * 1 = 12 and right side is 12 + 0 = 12, which satisfies the equation. This confirms that adding only 1 litre of paint gives the desired ratio.

Why Other Options Are Wrong:
Option B (1 litre water) would give 3 litres paint and 4 litres water, resulting in a ratio 3 : 4, which is not 4 : 3.
Option C (½ litre water and 1 litre paint) leads to 4 litres paint and 3.5 litres water, giving a ratio 4 : 3.5, which does not simplify to 4 : 3.
Option D (½ litre paint and 1 litre water) produces 3.5 litres paint and 4 litres water, ratio 3.5 : 4, again not equal to 4 : 3.

Common Pitfalls:
A common error is to treat 2 : 1.5 as 2 : 1 directly without simplifying correctly, which changes the target mixture. Another frequent mistake is to think equal additions of paint and water will preserve or create the required ratio, which is not true except in special cases. Properly simplifying the required ratio and systematically testing each option prevents these mistakes.

Final Answer:
The painter should add 1 litre of paint to obtain the correct proportion of paint to water.