1.A vessel is filled with liquid, 3 parts of which are water and 5 parts syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?
Question
1.A vessel is filled with liquid, 3 parts of which are water and 5 parts syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?
Solution
Let's solve this step by step:
Step 1: Let's assume the total volume of the mixture in the vessel is 1 unit (you can assume any unit: liter, gallon, etc. The unit doesn't matter because we're looking for a ratio). According to the problem, the vessel is filled with 3/8 units of water and 5/8 units of syrup.
Step 2: We need to find out how much of this mixture to remove and replace with water so that the mixture becomes half water and half syrup. Let's denote the volume of the mixture to be removed as 'x' units.
Step 3: When we remove 'x' units of the mixture, we're removing 'x' units of water and syrup in the ratio they exist in the vessel, which is 3:5. So, we remove 3x/8 units of water and 5x/8 units of syrup.
Step 4: We then replace this 'x' units with pure water. So, the volume of water in the vessel becomes 3/8 - 3x/8 + x = 1/2 (since we want the final mixture to be half water).
Step 5: Solving the equation 3/8 - 3x/8 + x = 1/2 for 'x' gives us x = 1/8 units.
So, we need to remove 1/8 of the total mixture and replace it with water to get a mixture that's half water and half syrup.
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