A chemist needs to mix a 5% saline solution with a 10% saline solution to make 2 liters of a 7% solution. How much of each is needed?
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Let's assume that the chemist needs x liters of the 5% saline solution and y liters of the 10% saline solution.
To make 2 liters of a 7% solution, the sum of the volumes of the 5% saline solution and the 10% saline solution must equal 2 liters:
x + y = 2 ----(1)
To determine the amounts of each solution needed, we need to express the concentration of the resulting solution in terms of the given concentrations.
The concentration of a solution can be calculated by taking the weighted average of the individual concentrations.
For instance, the resulting concentration of the 7% saline solution can be calculated using the following formula:
7 = (5x + 10y)/ (x + y) ----(2)
To solve this system of equations(1) and (2), we can multiply equation (1) by 5 and rewrite it as:
5x + 5y = 10 ----(3)
Now, we have a system of two equations:
5x + 5y = 10 ----(3)
5x + 10y = 14 ----(2)
Subtracting equation (3) from equation (2) gives:
(5x + 10y) - (5x + 5y) = 14 - 10
5y - 5y = 4
0 = 4
From the above equation, we can conclude that the system of equations has no solution.
Therefore, it is not possible to mix a 5% saline solution with a 10% saline solution to obtain a 2-liter solution with a concentration of 7%.
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