Question

Mixing a Solution A chemist wants to make 14 liters of a \(40 \%\) acid solution. She has solutions that are \(30 \%\) acid and \(65 \%\) acid. How much of each must she mix?

Short answer

Mix 10 liters of 30% acid solution with 4 liters of 65% acid solution.

Step-by-step solution

Define the Variables

Let x be the amount of 30% acid solution and y be the amount of 65% acid solution to be mixed.

Set Up the Volume Equation

The total volume of the mixture must be 14 liters. Therefore, we have the equation: x + y = 14.

Set Up the Concentration Equation

The total amount of acid in the solution must be 40% of 14 liters, which is 5.6 liters. Therefore, we have the equation: 0.30x + 0.65y = 5.6.

Solve the Volume Equation for One Variable

Solve the first equation for x: x = 14 - y.

Substitute into the Concentration Equation

Substitute x = 14 - y into the concentration equation: 0.30(14 - y) + 0.65y = 5.6.

Simplify the Equation

Distribute and combine like terms: 4.2 - 0.30y + 0.65y = 5.6 4.2 + 0.35y = 5.6.

Solve for y

Subtract 4.2 from both sides: 0.35y = 1.4 Divide both sides by 0.35: y = 4.

Find the Value of x

Substitute y = 4 back into the equation x = 14 - y: x = 14 - 4 x = 10.

Conclusion

The chemist needs to mix 10 liters of the 30% acid solution with 4 liters of the 65% acid solution.

Key concepts

linear equations

Linear equations are mathematical statements where the highest power of the variable is one. They look like any equation that can be written in the form: ax + b = c where 'a', 'b', and 'c' are constants. Linear equations are essential because they help us model relationships with a constant rate of change. In our problem, we set up a linear equation to represent total volume: x + y = 14. This shows the combined volume of two solutions. Solving it helps determine the quantities mixed to achieve our target.

system of equations

A system of equations is a set of two or more equations with the same variables. The solutions of the system satisfy all equations simultaneously. In the given exercise, we have the system: x + y = 14 0.30x + 0.65y = 5.6. Here, the first equation represents the total volume, and the second represents the concentration of acid in the mix. Solving such a system helps in finding precise amounts needed for each solution to match the given conditions.

concentration equations

Concentration equations describe how much of a substance (solute) there is within a specific amount of solution. For example, a 40% solution of acid means 40% of the total solution volume is acid. Our concentration equation for the exercise is: 0.30x + 0.65y = 5.6. This means the total acid from both solutions has to add up to 5.6 liters. Creating and solving concentration equations is crucial in chemistry and mixing problems to ensure desired properties of the final mix.

algebraic substitution

Algebraic substitution is a method used to solve systems of equations. It involves solving one equation for one variable and substituting this expression into the other equation. In our exercise, we first solve: x = 14 - y from the volume equation. We then substitute this into the concentration equation: 0.30(14 - y) + 0.65y = 5.6. This simplifies our system to one equation with one variable, making it easier to solve. Substitution helps break down complex problems into manageable steps, leading to the solution efficiently.