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.