A car radiator needs a 60% antifreeze solution. The radiator now holds 21 liters of a 40% solution. How many liters of this should be drained and replaced with 100% antifreeze to get the desired strength?
Let x be the number of liters drained and replaced with 100% antifreeze.
The amount of pure antifreeze in the 40% solution that will remain after x liters are drained is 0.4 * (21 - x) liters.
The amount of pure antifreeze in the 100% antifreeze that will be added is 1 * x liters.
The total amount of pure antifreeze in the new solution will be 0.4 * (21 - x) + x liters.
The total amount of solution after x liters are drained and replaced is 21 liters.
The concentration of antifreeze in the new solution is (0.4 * (21 - x) + x) / 21 = 0.6.
0.4 * (21 - x) + x = 0.6 * 21.
8.4 - 0.4x + x = 12.6.
0.6x = 12.6 - 8.4.
0.6x = 4.2.
x = 4.2 / 0.6.
x = 7.
The number of liters drained and replaced with 100% antifreeze is 7. Answer: \boxed{7}.
To solve this problem, we need to calculate the amount of 100% antifreeze that needs to be added and the amount of the 40% solution that needs to be drained.
Let's denote:
x = the amount of the 40% solution that needs to be drained (in liters)
y = the amount of 100% antifreeze that needs to be added (in liters)
The initial mixture contains 21 liters of a 40% solution. Therefore, the amount of actual antifreeze in the mixture is given by:
0.40 * 21 = 8.4 liters
After draining x liters of the initial mixture, the amount of actual antifreeze left in the mixture is:
8.4 - 0.40x liters
Now, when we add y liters of 100% antifreeze, the total amount of antifreeze in the mixture becomes:
8.4 - 0.40x + y liters
To achieve a 60% antifreeze solution, the total amount of the mixture should be:
21 liters
Therefore, we can set up the following equation:
8.4 - 0.40x + y = 0.60 * 21
Simplifying the equation, we get:
8.4 - 0.40x + y = 12.6
Rearranging the equation, we get:
0.40x - y = 12.6 - 8.4
0.40x - y = 4.2
Based on the given information, we know that the desired strength is 60% antifreeze. So, y liters of 100% antifreeze will also make up 60% of the final mixture. Therefore, y can be calculated as follows:
0.60 * (21 - x) = y
Substituting this value of y into the equation, we get:
0.40x - 0.60 * (21 - x) = 4.2
Now, we can solve the equation to find the value of x.
To solve this problem, we need to find out how many liters of the 40% solution should be drained and replaced with 100% antifreeze solution to get the desired 60% strength.
Let's break down the problem step by step:
Step 1: Determine the amount of antifreeze in the initial 40% solution.
In a 40% solution, 40% of the total volume is antifreeze. So, the initial solution contains (40/100) * 21 liters = 8.4 liters of antifreeze.
Step 2: Determine the amount of antifreeze needed to achieve the desired strength.
At a desired strength of 60%, the total volume of the solution remains the same, i.e., 21 liters. Therefore, the amount of antifreeze needed for the desired strength is (60/100) * 21 liters = 12.6 liters.
Step 3: Find the difference between the desired amount of antifreeze and the current amount of antifreeze.
The difference between the desired amount of antifreeze (12.6 liters) and the amount of antifreeze in the initial solution (8.4 liters) is 12.6 liters - 8.4 liters = 4.2 liters.
Step 4: Determine the ratio of the 40% solution that needs to be replaced.
Since we want to replace a certain amount of the 40% solution, the ratio of the solution to be replaced can be determined by taking the difference found above (4.2 liters) and dividing it by the concentration difference between the existing solution (40%) and the desired strength (60%).
Let's calculate this ratio:
Ratio = Difference in antifreeze amount / Difference in concentration
Ratio = 4.2 liters / (60% - 40%)
Ratio = 4.2 liters / 0.2
Ratio = 21 liters
Step 5: Calculate the amount of the 40% solution that needs to be replaced.
To determine the amount of the 40% solution to be replaced, multiply the ratio found above by the total volume of the initial solution:
Amount to be replaced = Ratio * Initial volume
Amount to be replaced = 21 liters * 21 liters
Amount to be replaced = 441 liters
Therefore, 441 liters of the 40% solution should be drained and replaced with 100% antifreeze to obtain the desired 60% strength.