Vessels P and Q contained 3 litres of milk and 3 litres of water respectively. The contents of P and Q were emptied into vessel R, which has a capacity of 7 litres. One-fifth of the mixture in R was replaced with milk. This process is repeated two more times. The milk concentration in R finally is _______%.
Question
Vessels P and Q contained 3 litres of milk and 3 litres of water respectively. The contents of P and Q were emptied into vessel R, which has a capacity of 7 litres. One-fifth of the mixture in R was replaced with milk. This process is repeated two more times. The milk concentration in R finally is _______%.
Solution
Step 1: Initially, vessel R contains 3 litres of milk and 3 litres of water, making a total of 6 litres.
Step 2: One-fifth of the mixture in R is replaced with milk. This means that one-fifth of the 6 litres, which is 1.2 litres, is removed and replaced with milk. Since the removed mixture is half milk and half water, 0.6 litres of milk and 0.6 litres of water are removed. After the replacement, the total amount of milk in R is 3 litres (initial) - 0.6 litres (removed) + 1.2 litres (added) = 3.6 litres. The total amount of water is 3 litres (initial) - 0.6 litres (removed) = 2.4 litres.
Step 3: The replacement process is repeated two more times. Each time, one-fifth of the mixture is removed and replaced with milk. The amount of milk and water removed each time is proportional to their current amounts in the mixture.
After the second replacement, the amount of milk is 3.6 litres - 0.23.6 litres (removed) + 1.2 litres (added) = 4.08 litres. The amount of water is 2.4 litres - 0.22.4 litres (removed) = 1.92 litres.
After the third replacement, the amount of milk is 4.08 litres - 0.24.08 litres (removed) + 1.2 litres (added) = 4.464 litres. The amount of water is 1.92 litres - 0.21.92 litres (removed) = 1.536 litres.
Step 4: The final concentration of milk in R is the amount of milk divided by the total amount of the mixture, which is 4.464 litres / 6 litres = 0.744 or 74.4%.
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