Solve each problem. See Example 3. Aryan wishes to strengthen a mixture from 10% alcohol to 30% alcohol. How much pure alcohol should be added to 7 L of the 10% mixture?

Verified step by step guidance

1. Identify the initial amount of alcohol in the 10% mixture: Calculate 10% of 7 L to find the amount of alcohol in the original mixture.
2. Let x be the amount of pure alcohol to be added. The pure alcohol is 100% alcohol.
3. Set up the equation for the final mixture: The total amount of alcohol after adding x liters of pure alcohol should be 30% of the total volume of the new mixture.
4. The equation will be: (0.10 * 7) + x = 0.30 * (7 + x).
5. Solve the equation for x to find out how much pure alcohol needs to be added.

Key Concepts

Concentration of Solutions

Concentration refers to the amount of solute (in this case, alcohol) present in a given volume of solution. It is often expressed as a percentage, indicating how much of the solution is made up of the solute. Understanding how to manipulate concentrations is crucial for solving mixture problems, as it allows one to determine how much solute needs to be added or removed to achieve a desired concentration.

Mixture Problems

Mixture problems involve combining two or more substances to achieve a specific concentration or quantity. These problems typically require setting up equations based on the initial and final concentrations and volumes of the mixtures. In this scenario, the goal is to find the volume of pure alcohol needed to increase the concentration of a 10% alcohol solution to 30%.

Algebraic Equations

Algebraic equations are mathematical statements that express the equality of two expressions. In the context of mixture problems, setting up an equation involves defining variables for unknown quantities and using known values to create a solvable equation. Solving these equations allows one to find the required amounts of substances to achieve the desired mixture concentration.