A pharmacist has an 18% alcohol solution and a 40% alcohol solution. How much of each should he mix together to make 10 liters of a 20% alcohol solution?
x * .18 + (10 - x) * .40 = 10 * .20
To solve this problem, we need to use the concept of mixture problems. Let's assume the pharmacist needs to mix x liters of the 18% alcohol solution and y liters of the 40% alcohol solution to make 10 liters of a 20% alcohol solution.
Now let's set up the equation based on the concentration of alcohol in the two solutions:
0.18x + 0.40y = 0.20(10)
Next, simplify the equation:
0.18x + 0.40y = 2
We also know that the total volume of the mixture should be 10 liters, so we have another equation:
x + y = 10
We now have a system of two equations:
0.18x + 0.40y = 2 ----------- (Equation 1)
x + y = 10 ------------------- (Equation 2)
To solve this system of equations, we can use the method of substitution or elimination.
Let's solve this system of equations using the method of substitution:
From Equation 2, we can express x in terms of y:
x = 10 - y
Now we substitute this expression for x into Equation 1:
0.18(10-y) + 0.40y = 2
1.8 - 0.18y + 0.40y = 2
Combine like terms:
0.22y = 0.2
Solving for y:
y = 0.2 / 0.22
y ≈ 0.909
Now substitute the value of y into Equation 2 to solve for x:
x + 0.909 = 10
x = 10 - 0.909
x ≈ 9.091
Therefore, the pharmacist needs to mix approximately 9.091 liters of the 18% alcohol solution with 0.909 liters of the 40% alcohol solution to obtain 10 liters of a 20% alcohol solution.
To solve this problem, we can use the concept of mixing solutions with different concentrations of a substance.
Let's assume that the pharmacist mixes x liters of the 18% alcohol solution and (10 - x) liters of the 40% alcohol solution to make 10 liters of a 20% alcohol solution.
Now, let's calculate the amount of alcohol in each solution and set up an equation using the concentration and volume of alcohol:
For the 18% alcohol solution:
Amount of alcohol = (18/100) * x
For the 40% alcohol solution:
Amount of alcohol = (40/100) * (10 - x)
Since we want to create a 20% alcohol solution, the total amount of alcohol in the mixture should be equal to:
Amount of alcohol = (20/100) * 10
Setting up the equation:
(18/100) * x + (40/100) * (10 - x) = (20/100) * 10
Simplifying the equation:
0.18x + 4 - 0.4x = 2
Combining like terms:
0.18x - 0.4x = 2 - 4
-0.22x = -2
x = -2 / -0.22
x ≈ 9.09
Therefore, the pharmacist should mix approximately 9.09 liters of the 18% alcohol solution and (10 - 9.09) ≈ 0.91 liters of the 40% alcohol solution to make 10 liters of a 20% alcohol solution.