How many gallons each of a 45% chlorine solution and a 30% chlorine solution should be mixed to obtained 24 gallons of a 35% chlorine solution?
amount of 45% stuff ---- x gall
amount of 30% stuff ---- 24-x gall
solve for x:
.45 + .3(24-x) = .35(24)
45 + 30(24-x) = 35(24)
...
To solve this problem, we can use the concept of the concentration of solutions. Let's assume that x represents the number of gallons of the 45% chlorine solution, and y represents the number of gallons of the 30% chlorine solution that we need to mix.
First, let's set up the equation based on the chlorine concentration:
(0.45x + 0.30y) / (x + y) = 0.35
This equation represents the average chlorine concentration of the mixed solution. We can simplify it by multiplying both sides by (x + y):
0.45x + 0.30y = 0.35(x + y)
Next, let's substitute the given values into the equation:
0.45x + 0.30y = 0.35(24)
Now, we can simplify the equation and solve for one variable:
0.45x + 0.30y = 8.4
45x + 30y = 840
To solve for two variables, we need another equation. In this case, we know that the total volume of the mixed solution is 24 gallons. Therefore:
x + y = 24
Now, we have a system of equations:
45x + 30y = 840
x + y = 24
We can use substitution or elimination method to solve this system of equations.
Let's use the elimination method. Multiply the second equation by 30 to eliminate y:
30(x + y) = 30(24)
30x + 30y = 720
Now, subtract this equation from the first equation:
45x + 30y - 30x - 30y = 840 - 720
15x = 120
Divide both sides by 15:
x = 120 / 15
x = 8
Now, substitute the value of x in the second equation:
8 + y = 24
y = 24 - 8
y = 16
Therefore, to obtain 24 gallons of a 35% chlorine solution, you should mix 8 gallons of a 45% chlorine solution with 16 gallons of a 30% chlorine solution.