Translate to a system of equations and solve. A scientist...

Translate to a system of equations and solve.
A scientist needs 65 liters of a $15 \%$ alcohol solution. She has available a $25 \%$ and a $12 \%$ solution. How many liters of the $25 \%$ and how many liters of the $12 \%$ solutions should she mix to make the $15 \%$ solution?

Key Concepts

Translating Word Problems into Equations

Translating word problems into mathematical equations is the process of identifying and defining variables, and then expressing the relationships and constraints described in the problem using algebra. This step is crucial in applying systematic methods, like solving systems of equations, to find the solution to the problem.

Percentage Concentrations

This concept involves understanding percentages as a way to express the concentration of a particular substance in a solution. In mixture problems, percentage concentrations are used to model the relative amount of a component in the solution, and algebraic equations can represent how these percentages combine when different solutions are mixed.

System of Equations

A system of equations involves multiple equations that share common variables. In many practical problems, such as mixing solutions, the relationships between quantities can be captured by a systems of equations which are then solved simultaneously to find the values of the variables that satisfy all the equations.

Mixture Problems

Mixture problems involve combining two or more components with different attributes, such as concentrations or percentages, to create a mixture with a desired overall attribute. The key challenge is to determine the appropriate amounts of each component so that the final mixture meets the specified criteria.

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