A container is full of milk. One-third of its contents is removed and replaced with water. The contents are mixed thoroughly before each removal. The same operation is repeated three more times. Find the final ratio of milk to water.
Answer and solution
Correct answer: B
Solution
Answer: B — 16:65
1. Assume the milk and water are uniformly mixed before each removal, as in standard replacement questions.
2. Each operation removes 1/3 of the milk currently present and retains 2/3. Replacing with water adds no milk.
3. Four operations leave (2/3)⁴=16/81 of the container as milk.
4. Water occupies 1−16/81=65/81. Hence milk: water=16:65, option B.
Why this method works:
Thorough mixing makes the fraction of milk removed equal to the fraction of the container removed. After each replacement the container is full again, so the same retention factor applies at every step rather than subtracting a fixed milk volume.
Independent check and common trap:
Taking capacity 81 units makes the sequence transparent: milk 81→54→36→24→16. The final water is 81−16=65. Four operations are required because the first operation is followed by three more; stopping after three would give 8:19.
Apply: One operation plus three MORE is four. Milk fraction=(2/3)⁴=16/81; water is its complement 65/81. Ratio 16:65.
Check: Taking capacity 81 units makes the sequence transparent: milk 81→54→36→24→16. The final water is 81−16=65. Four operations are required because the first operation is followed by three more; stopping after three would give 8:19.
1. Assume the milk and water are uniformly mixed before each removal, as in standard replacement questions.
2. Each operation removes 1/3 of the milk currently present and retains 2/3. Replacing with water adds no milk.
3. Four operations leave (2/3)⁴=16/81 of the container as milk.
4. Water occupies 1−16/81=65/81. Hence milk: water=16:65, option B.
Why this method works:
Thorough mixing makes the fraction of milk removed equal to the fraction of the container removed. After each replacement the container is full again, so the same retention factor applies at every step rather than subtracting a fixed milk volume.
Independent check and common trap:
Taking capacity 81 units makes the sequence transparent: milk 81→54→36→24→16. The final water is 81−16=65. Four operations are required because the first operation is followed by three more; stopping after three would give 8:19.
G Strategy shortcut
Spot: Mixtures and Alligation.Apply: One operation plus three MORE is four. Milk fraction=(2/3)⁴=16/81; water is its complement 65/81. Ratio 16:65.
Check: Taking capacity 81 units makes the sequence transparent: milk 81→54→36→24→16. The final water is 81−16=65. Four operations are required because the first operation is followed by three more; stopping after three would give 8:19.
G Strategy: Spot: Mixtures and Alligation. Apply: One operation plus three MORE is four. Milk fraction=(2/3)⁴=16/81; water is its complement 65/81. Ratio 16:65. Check: Taking capacity 81 units makes the sequence transparent: milk 81→54→36→24→16. The final water is 81−16=65. Four operations are required because the first operation is followed by three more; stopping after three would give 8:19.