A lab needs a 20 liters of a 15% acid solution. The lab only has a 10% acid solution and a 30% acid solution. How much of the 10% acid solution will they need to mix with the 30% solution to obtain 20 liters of a 15% solution?
To calculate the amount of the 10% acid solution needed to mix with the 30% acid solution, we can use the method of alligation.
Let's denote the amount of the 10% solution as x liters.
Let's start by creating a table to represent the solution mix:
Solution | % Acid | Liters
---------------------------------------------
10% Solution | 10% | x
30% Solution | 30% | (20 - x)
Now, let's determine the weighted average percentage of the acid in the final mix:
(10% * x) + (30% * (20 - x)) = 15% * 20
Simplifying the equation:
0.1x + 0.3(20 - x) = 0.15 * 20
0.1x + 6 - 0.3x = 3
-0.2x = -3
x = -3 / -0.2
x = 15
Therefore, the lab will need 15 liters of the 10% acid solution to mix with the 30% acid solution in order to obtain 20 liters of a 15% acid solution.
To solve this problem, we need to use the concept of a "mixture equation." Let's assume that x liters of the 10% acid solution needs to be mixed with y liters of the 30% acid solution to obtain 20 liters of a 15% acid solution.
Given:
Volume of the 10% acid solution = x liters
Volume of the 30% acid solution = y liters
Total volume of the mixture = 20 liters
Now, we can set up two equations based on the acid content and the total volume:
1. Acid Content Equation:
(0.10 * x) + (0.30 * y) = 0.15 * 20
The left side represents the total amount of acid in the mixture, which is the sum of the acid in the 10% and 30% solutions. The right side represents the acid content in the final 15% solution.
2. Volume Equation:
x + y = 20
Since the total volume of the mixture is 20 liters, the sum of individual volumes of the 10% and 30% solutions must be equal to 20.
Now, we have a system of two equations with two variables. We can solve this system using substitution or elimination.
Let's use the elimination method:
From the volume equation (x + y = 20), we can solve for x:
x = 20 - y
Substituting this value of x into the acid content equation:
(0.10 * (20 - y)) + (0.30 * y) = 0.15 * 20
Expand and simplify:
2 - 0.10y + 0.30y = 3
Combine like terms:
0.20y = 1
Divide by 0.20:
y = 5
Now, substituting the value of y back into the volume equation:
x + 5 = 20
x = 15
Therefore, the lab will need 15 liters of the 10% acid solution and 5 liters of the 30% acid solution to obtain 20 liters of a 15% solution.