GED Mathematical Reasoning TestAlgebraic Problem Solving with Expressions and EquationsHard
A chemist is mixing two solutions. Solution A is 20% acid, and Solution B is 50% acid. The chemist wants to produce 15 liters of a mixture that is 30% acid. How many liters of Solution A are needed?
- A10 liters
- B5 liters
- C12.5 liters
- D7.5 liters
Show answer & explanationAnswer & explanation
Correct answer: A. 10 liters
Let 'a' be the volume of Solution A and 'b' be the volume of Solution B. We have two equations: 1) a + b = 15 (total volume) and 2) 0.20a + 0.50b = 0.30(15) (total acid). From (1), b = 15 - a. Substitute into (2): 0.20a + 0.50(15 - a) = 4.5. Simplify: 0.20a + 7.5 - 0.50a = 4.5. Combine like terms: -0.30a + 7.5 = 4.5. Subtract 7.5: -0.30a = -3. Divide by -0.30: a = 10. So, 10 liters of Solution A are needed.
Why the other options are wrong
- B. This would result in (5*0.2 + 10*0.5)/15 = (1+5)/15 = 6/15 = 0.40 (40% acid), which is incorrect.
- C. This would result in (12.5*0.2 + 2.5*0.5)/15 = (2.5+1.25)/15 = 3.75/15 = 0.25 (25% acid), which is incorrect.
- D. This would result in (7.5*0.2 + 7.5*0.5)/15 = (1.5+3.75)/15 = 5.25/15 = 0.35 (35% acid), which is incorrect.
Systems of Linear Equations (Mixture Problems)
Using two or more linear equations to solve problems involving the combination of two or more substances with different concentrations to achieve a desired mixture.
- One equation typically represents the total quantity of the mixture.
- Another equation represents the total amount of a specific component (e.g., acid, sugar) in the mixture.
- Can be solved using substitution or elimination methods.
Memory trick: Mixture: Total volume, total component!