Solve each problem. See Example 3. How many gallons of a 5% acid solution must be mixed with 5 gal of a 10% solution to obtain a 7% solution?

Verified step by step guidance

1. Define the variable: Let \(x\) represent the number of gallons of the 5% acid solution to be mixed.
2. Set up the equation based on the amount of pure acid in each solution. The total amount of acid from the 5% solution is \$0.05x\(, and from the 10% solution is \)0.10 \times 5$ gallons.
3. The total volume of the mixture is \((x + 5)\) gallons, and the desired concentration is 7%, so the amount of acid in the mixture is \$0.07(x + 5)$.
4. Write the equation representing the total acid from both solutions equal to the acid in the mixture: \$0.05x + 0.10 \times 5 = 0.07(x + 5)$.
5. Solve the equation for \(x\) by first expanding and simplifying both sides, then isolating \(x\) to find the number of gallons of the 5% solution needed.

Key Concepts

Mixture Problems

Mixture problems involve combining two or more solutions with different concentrations to form a new solution with a desired concentration. The key is to set up an equation based on the total amount of substance (e.g., acid) before and after mixing.

Concentration Percentage

Concentration percentage represents the amount of solute (acid) per total solution volume, expressed as a percent. For example, a 5% acid solution means 5 gallons of acid per 100 gallons of solution, which helps in calculating the amount of acid in each solution.

Setting Up and Solving Linear Equations

To find the unknown volume, set up a linear equation equating the total amount of acid before and after mixing. Solving this equation involves basic algebraic manipulation to isolate the variable representing the unknown quantity.