A beaker contained V litres of a mixture of milk and water, with milk and water in the ratio of 3 : 2. The total volume of the mixture was increased by 60% by adding water. Next, 38.4 litres of the solution in the beaker was replaced by water. If the final ratio of milk and water in the beaker is 3 : 7, find the value of V (in litres).8096120192
Question
A beaker contained V litres of a mixture of milk and water, with milk and water in the ratio of 3 : 2. The total volume of the mixture was increased by 60% by adding water. Next, 38.4 litres of the solution in the beaker was replaced by water. If the final ratio of milk and water in the beaker is 3 : 7, find the value of V (in litres).8096120192
Solution
Step 1: Let's denote the initial volume of milk as M and the initial volume of water as W. Given that the ratio of milk to water is 3:2, we can write the following equations:
M = 3x W = 2x
Step 2: The total volume of the mixture is V, so we have:
V = M + W V = 3x + 2x V = 5x
Step 3: The total volume of the mixture was increased by 60% by adding water. So, the new volume of the mixture is 1.6V and the volume of water added is 0.6V. Therefore, the new volume of water is W + 0.6V = 2x + 0.6*5x = 5x.
Step 4: Then, 38.4 litres of the solution in the beaker was replaced by water. The volume of milk remains the same, but the volume of water increases by 38.4 litres. So, the new volume of water is 5x + 38.4.
Step 5: The final ratio of milk to water in the beaker is 3:7. So, we can write the following equation:
M / (5x + 38.4) = 3 / 7
Step 6: Solving the above equation, we get:
3x = (3/7) * (5x + 38.4) 21x = 15x + 115.2 6x = 115.2 x = 19.2
Step 7: Substituting x = 19.2 into the equation V = 5x, we get:
V = 5 * 19.2 V = 96 litres
So, the initial volume of the mixture in the beaker was 96 litres.
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