A car radiator needs a 40% antifreeze solution. The radiator now holds 18 liters of a 10% solution. How many liters of this should be drained and replaced with 100% antifreeze to get the desired strength?
Let's start by finding the amount of antifreeze currently in the radiator.
At present, the radiator holds 18 liters of a 10% antifreeze solution, so the amount of antifreeze in the radiator is 18 liters * 10% = 1.8 liters.
Now, let's determine how much antifreeze needs to be in the radiator for a 40% concentration.
If we let x represent the amount of the 10% solution to be drained, then we can set up the equation:
(1.8 liters + x) = 0.4 * (18 liters - x)
Expanding this equation, we get:
1.8 liters + x = 7.2 liters - 0.4x
Combining like terms, we have:
1.8 liters + 0.4x = 7.2 liters - x
Now, let's solve for x:
1.4x = 5.4 liters
Dividing both sides by 1.4, we find:
x = 3.857 liters
Therefore, approximately 3.857 liters of the 10% solution should be drained and replaced with 100% antifreeze to get the desired 40% strength.
To find the number of liters of the 10% solution that should be drained and replaced with 100% antifreeze, you can follow these steps:
Step 1: Find the amount of antifreeze in the current solution.
The current solution has 18 liters of a 10% antifreeze solution.
Amount of antifreeze = 18 liters * 0.10 = 1.8 liters
Step 2: Determine the total volume of the new solution.
Let x be the number of liters drained and replaced with 100% antifreeze.
The new solution will have a total volume of 18 liters.
Step 3: Find the amount of antifreeze in the final solution.
After draining and replacing x liters of the 10% solution, the amount of antifreeze is still 1.8 liters.
The amount of antifreeze in the new solution is:
Amount of antifreeze = (1.8 liters - 0.10x) + (x liters * 1)
Step 4: Set up an equation and solve for x.
The final solution needs to be a 40% antifreeze solution.
So, the amount of antifreeze in the final solution should be 40% of the total volume:
Amount of antifreeze = 0.40 * 18 liters
Setting up the equation:
(1.8 liters - 0.10x) + x liters = 0.40 * 18 liters
Simplifying the equation:
1.8 liters - 0.10x + x liters = 7.2 liters
Combining like terms:
0.9x = 7.2 liters - 1.8 liters
0.9x = 5.4 liters
Solving for x:
x = 5.4 liters / 0.9
x ≈ 6 liters
Therefore, approximately 6 liters of the 10% solution should be drained and replaced with 100% antifreeze to get the desired 40% strength.
To solve this problem, we need to find out how many liters of the 10% solution need to be drained and replaced with 100% antifreeze to achieve a 40% solution.
Let's break down the problem step by step:
Step 1: Determine the initial amount of antifreeze in the 10% solution.
The 10% solution contains 10% antifreeze. Therefore, we can calculate the initial amount of antifreeze as 10% of 18 liters:
Initial antifreeze = 0.10 * 18 liters = 1.8 liters
Step 2: Determine the desired amount of antifreeze in the final solution.
We want the final solution to have a concentration of 40% antifreeze. To determine the desired amount of antifreeze, we need to calculate 40% of the total volume of the final solution.
Desired antifreeze = 0.40 * (18 liters - x) liters
Step 3: Set up the equation.
To find the amount of the 10% solution that should be drained and replaced with 100% antifreeze, we need to set up an equation stating that the initial antifreeze plus the additional antifreeze added is equal to the desired antifreeze. Mathematically, this can be written as:
Initial antifreeze + Additional antifreeze = Desired antifreeze
Step 4: Solve the equation.
Substituting the values, we get:
1.8 liters + 1 (liters added) = 0.40 * (18 liters - x) liters
Simplifying the equation gives us:
1.8 liters + 1 liter = 7.2 liters - 0.40x
2.8 liters = 7.2 liters - 0.40x
Next, we will isolate the variable on one side of the equation:
0.40x = 7.2 liters - 2.8 liters
0.40x = 4.4 liters
Finally, we can solve for x by dividing both sides of the equation by 0.40:
x = 4.4 liters / 0.40
x = 11 liters
Therefore, you need to drain and replace 11 liters of the 10% solution with 100% antifreeze to obtain the desired 40% strength.