a radiator contains 25 quarts of a freeze water and anti freeze solution ,of which 60% anti freeze. how much of this solution should be drained and replace with water for the new solution to be 40% anti freeze?
To solve this problem, we need to determine the amount of the solution that should be drained and replaced with water to achieve a desired concentration. Let's break down the steps:
Step 1: Identify the given information.
- The radiator contains 25 quarts of a freeze water and antifreeze solution.
- The current solution has a concentration of 60% antifreeze.
Step 2: Define variables.
- Let's assume that x represents the amount of solution to be drained and replaced with water (in quarts).
Step 3: Set up the equation based on the given information.
- The amount of antifreeze in the original solution is 60% of 25 quarts, which is calculated as 0.6 * 25 = 15 quarts (since 60% is equal to 0.6).
- After draining x quarts, the remaining amount of antifreeze will be 15 - (0.6x), as 0.6x quarts of antifreeze is being removed.
Step 4: Set up the equation to achieve the desired concentration.
- The final amount of antifreeze in the new solution will be 40% of the total volume (25 - x) quarts, calculated as 0.4 * (25 - x).
- So, the equation becomes 15 - 0.6x = 0.4 * (25 - x).
Step 5: Solve the equation.
- Solve the equation from the previous step to find x:
15 - 0.6x = 10 - 0.4x (distributed 0.4 to terms inside parentheses)
-0.6x + 0.4x = 10 - 15 (combined like terms)
-0.2x = -5 (simplified)
x = -5 / (-0.2) (divided by -0.2)
x = 25 (quarts) (divided and simplified)
Step 6: Check the answer.
- Since we obtained a positive value (25 quarts) for x, we can conclude that draining 25 quarts of the original solution and replacing it with 25 quarts of water will result in a new solution with a 40% antifreeze concentration.
In summary, you would need to drain 25 quarts of the original solution and replace it with 25 quarts of water for the new solution to have a 40% antifreeze concentration.