A radiator contains 6 liters of a 25% antifreeze solution. How much should be drained and replaced with pure antifreeze to produce a 33% antifreeze solution?
If x is drained, then
.25(6-x) + 1.00(x) = .33(6)
Well, to produce a 33% antifreeze solution, you'll need to employ some radiator math and a dash of clownery! Let's get to it.
To figure out the quantity of antifreeze that needs to be drained, let's call it "X." Now, since the radiator contains 6 liters of a 25% antifreeze solution, we have 6 * 0.25 = 1.5 liters of pure antifreeze in it initially.
After draining out "X" liters of the solution, we will have 6 - X liters of the original solution left. According to our plans, we will now replace this amount with pure antifreeze, ensuring the final solution is 33% antifreeze.
Now, if we mix 1.5 liters of pure antifreeze with X liters of the original solution, it should be equal to (6 - X) liters of the original solution at 33% antifreeze, right?
So, (1.5 + X) * 1 = (6 - X) * 0.33
We can simplify this equation to 1.5 + X = 1.98 - 0.33X
By rearranging, we get:
1.33X = 0.48
And solving for X:
X ≈ 0.36 liters
So, to create a 33% antifreeze solution, you should drain approximately 0.36 liters from the radiator and replace it with an equal amount of pure antifreeze! Just be sure not to let any clowns spill it along the way!
To solve this problem, we can follow these steps:
Step 1: Calculate the amount of antifreeze in the initial solution.
Since the solution contains 25% antifreeze, we know that 25% of 6 liters is antifreeze.
Amount of antifreeze in the initial solution = 6 liters × (25/100) = 1.5 liters.
Step 2: Set up the equation.
Let's assume x liters of the 25% antifreeze solution is drained and replaced with pure antifreeze.
The amount of antifreeze drained from the initial solution would then be 1.5 liters × (x/6) since the total amount in the solution has decreased by x liters.
The amount of pure antifreeze added would be x liters.
Step 3: Calculate the amount of antifreeze in the final solution.
After replacing x liters of the 25% antifreeze solution with x liters of pure antifreeze, the total amount of antifreeze in the final solution would be 1.5 liters - (1.5 liters × (x/6)) + x liters.
Step 4: Set up the equation for the final concentration.
The concentration of antifreeze is defined as the ratio of the amount of antifreeze to the total volume of the solution.
For the final solution, antifreeze concentration = (1.5 - 1.5x/6 + x)/(6 - x).
Since we want the final concentration to be 33% (or 33/100), we can set up the equation as follows:
(1.5 - 1.5x/6 + x)/(6 - x) = 33/100.
Step 5: Solve for x.
To solve the equation, we can cross-multiply:
(1.5 - 1.5x/6 + x)(6 - x) = (33/100)(6 - x).
Multiplying both sides of the equation will give us a quadratic equation. Simplify and rearrange to get:
15 - 1.5x + 6x - x^2 = (33/100)(6 - x).
100(15 - 1.5x + 6x - x^2) = 33(6 - x).
1500 - 150x + 600x - 100x^2 = 198x - 33x^2.
33x^2 + 132x - 198x + 100x^2 + 198x - 1500 = 0.
133x^2 - 1500 = 0.
Now, we can solve this equation to find the value of x.
To determine the solution, we need to find out how much of the 25% antifreeze solution should be drained and replaced with pure antifreeze to achieve a 33% antifreeze solution in the radiator.
Let's break down the problem step-by-step:
Step 1: Identify the given information:
- The radiator contains 6 liters of a 25% antifreeze solution.
Step 2: Set up the equation:
Let x be the amount (in liters) of the 25% antifreeze solution to be drained and replaced with pure antifreeze.
Step 3: Determine the initial amount of antifreeze in the radiator:
The initial amount of antifreeze in the radiator can be calculated as:
(initial amount of antifreeze) = (initial volume of solution) * (concentration of antifreeze)
Given that the initial volume of solution is 6 liters, and the concentration of antifreeze is 25%, we have:
(initial amount of antifreeze) = 6 * 25/100 = 1.5 liters
Step 4: Determine the final amount of antifreeze in the radiator:
The final amount of antifreeze in the radiator can be calculated as:
(final amount of antifreeze) = (final volume of solution) * (concentration of antifreeze)
Since we want to achieve a 33% antifreeze solution in the radiator, the final volume of solution will be 6 liters (as it is not changed). Therefore, we have:
(final amount of antifreeze) = 6 * 33/100 = 1.98 liters (approximately)
Step 5: Set up the equation for the amount of antifreeze:
Since x liters of the 25% antifreeze solution is drained and replaced, the equation becomes:
1.5 - x + x = 1.98
Simplifying the equation:
1.5 - x + x = 1.98
1.5 = 1.98
x = 0.48 liters
Step 6: Determine the amount of solution to be drained:
The amount of 25% antifreeze solution to be drained is x = 0.48 liters.
Therefore, approximately 0.48 liters of the 25% antifreeze solution should be drained, and pure antifreeze should be added to the radiator to achieve a 33% antifreeze solution.
Note: It's always best to double-check the calculations and verify the answer with the appropriate units.