A chemist wishes to make 10 gallons of a 15 % acid solution...

A chemist wishes to make 10 gallons of a $15 \%$ acid solution by mixing a $10 \%$ acid solution with a $25 \%$ acid solution. (a) Let $x$ and $y$ denote the total volumes (in gallons) of the $10 \%$ and $25 \%$ solutions, respectively. Using the variables $x$ and $y,$ write an equation for the total volume of the $15 \%$ solution (the mixture). (b) Using the variables $x$ and $y,$ write an equation for the total volume of acid in the mixture by noting that Volume of acid in $15 \%$ solution $=$ volume of acid in $10 \%$ solution $+$ volume of acid in $25 \%$ solution. (c) Solve the system of equations from parts (a) and (b), and interpret your solution. (d) Is it possible to obtain a $5 \%$ acid solution by mixing a $10 \%$ solution with a $25 \%$ solution? Explain without solving any equations.

A chemist wishes to make 10 gallons of a $15 \%$ acid solution by mixing a $10 \%$ acid solution with a $25 \%$ acid solution.
(a) Let $x$ and $y$ denote the total volumes (in gallons) of the $10 \%$ and $25 \%$ solutions, respectively. Using the variables $x$ and $y,$ write an equation for the total volume of the $15 \%$ solution (the mixture).
(b) Using the variables $x$ and $y,$ write an equation for the total volume of acid in the mixture by noting that
Volume of acid in $15 \%$ solution $=$ volume of acid in $10 \%$ solution $+$ volume of acid in $25 \%$ solution.
(c) Solve the system of equations from parts (a) and (b), and interpret your solution.
(d) Is it possible to obtain a $5 \%$ acid solution by mixing a $10 \%$ solution with a $25 \%$ solution? Explain without solving any equations.

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Key Concepts

Mixture Problems

This concept involves combining solutions or mixtures of different concentrations to achieve a final solution with a desired concentration. The key idea is to determine how much of each component is required, taking into consideration the concentrations and total volumes involved in the process.

System of Linear Equations

Mixture problems typically lead to the formation of a system of linear equations where one equation represents the total volume constraint and the other represents the total amount of the solute (or acid) present. Solving these equations simultaneously provides the amount of each component needed in the mixture.

Weighted Average and Concentration

The concentration of a mixture is a weighted average based on the concentrations of the individual components and their volumes. This concept emphasizes that the final concentration is obtained by distributing the total amount of solute over the total volume of the mixture.

Feasibility Analysis in Mixture Problems

This involves determining whether a specified final concentration can be achieved by mixing solutions with given concentrations. It entails checking if the desired concentration lies between the concentrations of the available solutions, ensuring that the mixing process is physically and chemically possible.

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