A pharmacist has 40% and 60% iodine solutions on hand. How many liters of each iodine solution will be required to produce 4 liters of a 50% iodine.
.4x + .6(5-x) = .5(4)
x=5
so, 5L of 40% and 5L of 60%
makes sense, since 50% is halfway between the two.
5, 6
To find out how many liters of each iodine solution are needed, we can set up two equations based on the given information.
Let's denote:
x = liters of 40% iodine solution
y = liters of 60% iodine solution
According to the equation for the amount of iodine in the solution:
0.40x + 0.60y = 0.50 * 4
Simplifying this equation, we get:
0.40x + 0.60y = 2
Now, we need to establish a second equation based on the total volume of the solutions:
x + y = 4
We have a system of equations:
0.40x + 0.60y = 2
x + y = 4
We can solve this system of equations using the method of substitution or elimination.
Let's use the substitution method:
Rearrange the second equation as x = 4 - y
Substituting this value of x into the first equation:
0.40(4 - y) + 0.60y = 2
Simplifying the equation:
1.6 - 0.40y + 0.60y = 2
0.20y = 0.4
y = 0.4 / 0.20
y = 2
Now substitute this value of y back into the rearranged second equation:
x + 2 = 4
x = 4 - 2
x = 2
Therefore, to produce 4 liters of a 50% iodine solution, you will need 2 liters of 40% iodine solution and 2 liters of 60% iodine solution.
To find the number of liters of each iodine solution required to produce a 50% iodine solution, we can use a method called the "mixture" or "alligation" method.
Let's assume we need x liters of the 40% iodine solution and (4 - x) liters of the 60% iodine solution to produce a total of 4 liters of a 50% iodine solution.
Now, let's calculate the iodine content in each solution:
For the 40% iodine solution:
Iodine content = 40% of x liters = (40/100) * x
For the 60% iodine solution:
Iodine content = 60% of (4 - x) liters = (60/100) * (4 - x)
Since we want to produce a 50% iodine solution, the total iodine content in both solutions should be equal to 50% of 4 liters, which is (50/100) * 4.
Setting up the equation:
(40/100) * x + (60/100) * (4 - x) = (50/100) * 4
Now, let's solve for x:
(40/100) * x + (60/100) * (4 - x) = (50/100) * 4
(40/100) * x + (240/100) - (60/100) * x = 2
(40x + 240 - 60x) / 100 = 2
(240 - 20x) / 100 = 2
240 - 20x = 200
-20x = 200 - 240
-20x = -40
x = -40 / -20
x = 2
Therefore, you will need 2 liters of the 40% iodine solution and (4 - 2) = 2 liters of the 60% iodine solution to produce 4 liters of a 50% iodine solution.