A chemist needs to mix an 18% acid solution with a 45% acid solution to obtain a 12 liter mixture consisting of 36% acid. How many liters of each acid solutions must be used?
To solve this problem, we need to set up a system of equations based on the information given.
Let's assume that the chemist needs to mix x liters of the 18% acid solution and y liters of the 45% acid solution.
Given:
Total volume of the mixture = 12 liters
Concentration of acid in the mixture = 36%
Based on the given information, we can write the following equations:
Equation 1: x + y = 12 (to account for the total volume of the mixture)
Equation 2: (0.18x + 0.45y) / 12 = 0.36 (to account for the concentration of acid in the mixture)
Now, we can solve this system of equations.
Step 1: Solve Equation 1 for either x or y. Let's solve for y:
y = 12 - x
Step 2: Substitute the value of y in Equation 2:
(0.18x + 0.45(12 - x)) / 12 = 0.36
Step 3: Simplify the equation:
0.18x + 0.45(12 - x) = 0.36 * 12
0.18x + 5.4 - 0.45x = 4.32
Step 4: Combine like terms:
0.18x - 0.45x = 4.32 - 5.4
-0.27x = -1.08
Step 5: Divide both sides of the equation by -0.27 to solve for x:
x = -1.08 / -0.27
x = 4
Step 6: Substitute the value of x back into Equation 1 to find y:
4 + y = 12
y = 12 - 4
y = 8
Therefore, the chemist needs to mix 4 liters of the 18% acid solution and 8 liters of the 45% acid solution to obtain a 12 liter mixture consisting of 36% acid.
To solve this problem, we can use a technique called the method of mixtures.
Let's denote the volume of the 18% acid solution as A and the volume of the 45% acid solution as B.
Since we want to obtain a 12 liter mixture, we can set up the equation:
A + B = 12 (Equation 1)
Now, let's consider the amount of acid in each solution. The 18% acid solution contains 18% acid, which means it contains 0.18A liters of acid. Similarly, the 45% acid solution contains 45% acid, or 0.45B liters of acid.
Since the final mixture has a concentration of 36% acid, it contains 0.36 * 12 = 4.32 liters of acid.
Now we can set up another equation based on the amount of acid:
0.18A + 0.45B = 4.32 (Equation 2)
We now have a system of equations with two variables (A and B). We can solve this system simultaneously.
One way to solve this is by substitution. We can rearrange Equation 1 to express A in terms of B:
A = 12 - B
We substitute this expression for A in Equation 2:
0.18(12 - B) + 0.45B = 4.32
Now we solve for B:
2.16 - 0.18B + 0.45B = 4.32
Combining like terms:
0.27B = 2.16
Dividing both sides by 0.27:
B = 8
Now we can substitute this value of B into Equation 1 to find A:
A = 12 - 8 = 4
Therefore, the chemist should mix 4 liters of the 18% acid solution with 8 liters of the 45% acid solution to obtain a 12 liter mixture consisting of 36% acid.
amount of 18% solution --- x L
amount of 45% solution --- 12-x L
.18x + .45(12-x) = .36(12)
times 100
18x + 45(12-x) = 36(12)
take it from here