There is a vessel whose 80% volume is filled with milk and water solution. The concentration of milk in the solution is 90%. Using a steel mug, one mug-full of solution is withdrawn and replaced by an equal quantity of water. After 3 such replacements, the concentration of milk in the solution becomes 30.87%. What is the ratio of volumes of the vessel and the steel mug?
Question
There is a vessel whose 80% volume is filled with milk and water solution. The concentration of milk in the solution is 90%. Using a steel mug, one mug-full of solution is withdrawn and replaced by an equal quantity of water. After 3 such replacements, the concentration of milk in the solution becomes 30.87%. What is the ratio of volumes of the vessel and the steel mug?
Solution
Let's denote the volume of the vessel as V and the volume of the mug as M.
Step 1: Initially, the vessel is filled to 80% of its volume with a milk and water solution, where milk makes up 90% of the solution. Therefore, the volume of milk in the vessel is 0.8V * 0.9 = 0.72V.
Step 2: After each replacement, the volume of the milk in the vessel decreases by the volume of the milk in the mug, which is 90% of the mug's volume. Therefore, after 3 replacements, the volume of the milk in the vessel is 0.72V * (1 - 0.9)^3 = 0.72V * 0.001 = 0.00072V.
Step 3: After 3 replacements, the concentration of milk in the solution is 30.87%, which means that the volume of milk in the vessel is 0.3087 * 0.8V = 0.24696V.
Step 4: By equating the two expressions for the volume of milk in the vessel after 3 replacements, we get 0.00072V = 0.24696V. Solving this equation for V, we get V = 0.24696 / 0.00072 = 343.27778.
Step 5: The ratio of the volumes of the vessel and the mug is V/M = 343.27778. Therefore, the ratio of the volumes of the vessel and the mug is approximately 343:1.
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