Digital SATMath: AlgebraHard
A chemist is mixing two solutions. Solution A is 10% acid, and Solution B is 40% acid. How many liters of Solution B must be mixed with 5 liters of Solution A to obtain a mixture that is 25% acid?
- A4 liters
- B5 liters
- C6 liters
- D3 liters
Show answer & explanationAnswer & explanation
Correct answer: B. 5 liters
Let x be the liters of Solution B. The total amount of acid from Solution A is 0.10 * 5 = 0.5 liters. The total amount of acid from Solution B is 0.40 * x liters. The total volume of the mixture is (5 + x) liters. The total amount of acid in the mixture is 0.25 * (5 + x). Set up the equation: 0.5 + 0.4x = 0.25(5 + x). Solve for x: 0.5 + 0.4x = 1.25 + 0.25x => 0.15x = 0.75 => x = 5. So, 5 liters of Solution B are needed.
Why the other options are wrong
- A. Incorrect. If 4 liters of B are used, the total acid is 0.5 + 0.4(4) = 2.1. Total volume 9. 2.1/9 = 0.233 (23.3%).
- C. Incorrect. If 6 liters of B are used, the total acid is 0.5 + 0.4(6) = 2.9. Total volume 11. 2.9/11 = 0.2636 (26.36%).
- D. Incorrect. If 3 liters of B are used, the total acid is 0.5 + 0.4(3) = 1.7. Total volume 8. 1.7/8 = 0.2125 (21.25%).
Mixture Problems
Algebraic problems involving combining two or more quantities with different concentrations or values to achieve a desired overall concentration or value.
- Often solved using systems of linear equations or a single variable equation.
- Key is to track the total amount of a specific component (e.g., acid, solute).
- Equation typically balances the sum of component amounts with the desired total amount.
Memory trick: Mix the parts, balance the whole, percentage is key to the goal.