A chemist wants to create a solution that is 24% acid. How many liters of 40% acid solution must be added to 50 liters of an 18% acid solution to obtain this 24% acid mixture?
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To find the number of liters of the 40% acid solution needed to obtain the desired 24% acid mixture, follow these steps:
Step 1: Set up the equation using the given information:
Let "x" represent the number of liters of the 40% acid solution.
The initial acid amount from the 40% acid solution is 40% of x liters, which is 0.4x.
The initial acid amount from the 18% acid solution is 18% of 50 liters, which is 0.18 * 50 = 9.
The total acid amount after mixing is 24% of (x + 50) liters, which is 0.24(x + 50).
Step 2: Set up the equation using the acid amounts:
The equation becomes:
0.4x + 9 = 0.24(x + 50)
Step 3: Solve the equation:
Distribute 0.24 on the right side of the equation:
0.4x + 9 = 0.24x + 12
Combine like terms:
0.4x - 0.24x = 12 - 9
0.16x = 3
Divide by 0.16:
x = 3 / 0.16
Step 4: Calculate the value of x:
x ≈ 18.75
Therefore, approximately 18.75 liters of the 40% acid solution must be added to 50 liters of the 18% acid solution to obtain a 24% acid mixture.
To solve this problem, we need to calculate the amount of the 40% acid solution required.
Let's break down the problem:
1. The chemist has 50 liters of an 18% acid solution.
2. The chemist wants to create a solution that is 24% acid.
3. The 40% acid solution needs to be added to the existing 50 liters of 18% acid solution.
To determine the amount of 40% acid solution needed, follow these steps:
Step 1: Define variables
Let's assume the amount of 40% acid solution needed is "x" liters.
Step 2: Calculate acid in the 18% solution
The acid in the 18% solution can be calculated using the formula: Acid = (18/100) * 50.
Step 3: Calculate acid in the 40% solution to achieve desired mixture
The acid in the 40% solution can be calculated using the formula: Acid = (40/100) * x.
Step 4: Calculate acid in the desired mixture
The acid in the desired mixture can be calculated using the formula: Acid = (24/100) * (50 + x).
Step 5: Set up the equation
Since the acid in the desired mixture comes from both the 18% solution and the 40% solution, we can write the equation:
(18/100) * 50 + (40/100) * x = (24/100) * (50 + x).
Step 6: Solve the equation
Now we can solve the equation from Step 5 to find the value of x, which represents the amount of 40% acid solution needed.
Multiply through the equation to eliminate the fractions:
(18/100) * 50 + (40/100) * x = (24/100) * (50 + x).
(9/50) * 50 + (2/5) * x = (6/25) * (50 + x).
Simplify the equation:
9 + (2/5) * x = (6/25) * (50 + x).
9 + (2/5) * x = (3/5) * (10 + x).
Multiply both sides by 5 to eliminate the fractions:
45 + 2x = 3(10 + x).
45 + 2x = 30 + 3x.
Rearrange the equation:
2x - 3x = 30 - 45.
-x = -15.
Multiply both sides by -1 to solve for x:
x = 15.
Step 7: Find the answer
Therefore, the chemist needs to add 15 liters of the 40% acid solution to the 50 liters of the 18% acid solution to obtain a 24% acid mixture.