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 known quantities: the initial volume of the mixture is 7 liters, and it contains 10% alcohol. This means the amount of alcohol initially is \(7 \times 0.10\) liters.
2. Let \(x\) be the amount of pure alcohol (100% alcohol) to be added. Since pure alcohol is 100%, the amount of alcohol added is simply \(x\) liters.
3. After adding \(x\) liters of pure alcohol, the total volume of the mixture becomes \(7 + x\) liters, and the total amount of alcohol becomes the initial alcohol plus the added alcohol, which is \(7 \times 0.10 + x\) liters.
4. Set up an equation to represent the final concentration of alcohol as 30%. The concentration is the amount of alcohol divided by the total volume, so write: \(\frac{7 \times 0.10 + x}{7 + x} = 0.30\).
5. Solve the equation for \(x\) by multiplying both sides by \((7 + x)\), expanding, and isolating \(x\) on one side to find how much pure alcohol should be added.

Key Concepts

Concentration and Percentage Solutions

Concentration refers to the amount of a substance (like alcohol) present in a mixture, often expressed as a percentage. Understanding how to interpret and manipulate these percentages is essential for solving mixture problems involving solutions.

Setting Up and Solving Linear Equations

Mixture problems typically require forming an equation based on the total amount and concentration before and after adding a substance. Solving this linear equation helps find the unknown quantity, such as the volume of pure alcohol to add.

Conservation of Quantity in Mixtures

This concept involves recognizing that the total amount of the substance (alcohol) changes only by the amount added, while the total volume changes accordingly. Balancing these quantities ensures the final concentration matches the desired percentage.