Problem 60 - Algebra 2

Question

Vince needs 12 quarts of a 60% anti-freeze solution. He will combine an amount of 100% anti-freeze with an amount of a 50% anti-freeze solution. How many quarts of each solution should be mixed to make the required amount of the 60% anti-freeze solution?

Step-by-Step Solution

Answer

Mix 2.4 quarts of 100% anti-freeze with 9.6 quarts of 50% anti-freeze.

Step 1: Define Variables

Let \( x \) be the quarts of 100% anti-freeze solution and \( y \) be the quarts of 50% anti-freeze solution. We need to find the values of \( x \) and \( y \).

Step 2: Set Up Equations Based on Total Volume

The total volume of the solution should be 12 quarts. Hence, the equation is: \[ x + y = 12 \]

Step 3: Set Up Equations Based on Concentration

The final solution has a concentration of 60%. Therefore, the second equation based on concentration is: \[ 1.0x + 0.5y = 0.6 \times 12 \] Simplifying, we have: \[ x + 0.5y = 7.2 \]

Step 4: Solve the System of Equations

From the equations \( x + y = 12 \) and \( x + 0.5y = 7.2 \), we solve for \( x \) and \( y \). Subtract the second equation from the first: \[ (x + y) - (x + 0.5y) = 12 - 7.2 \] which simplifies to: \[ 0.5y = 4.8 \] So, \( y = 9.6 \). Substitute \( y = 9.6 \) into \( x + y = 12 \): \[ x + 9.6 = 12 \] Therefore, \( x = 2.4 \).

Step 5: Verify Solution

Verify by plugging \( x = 2.4 \) and \( y = 9.6 \) into the concentration equation: \[ x + 0.5y = 7.2 \]\[ 2.4 + 0.5(9.6) = 7.2 \] This equation holds true, confirming the solution is correct.

Key Concepts

Solution Concentration

Solution concentration describes how much solute is present in a given volume of solution. In our exercise, we are dealing with antifreeze solution concentrations. The initial solutions have different concentrations: one is 100% and another is 50% antifreeze. Our goal is to achieve a final concentration of 60% antifreeze. For this problem, Vince needs to mix these two different solutions to get a specific concentration in a total of 12 quarts.

• The 100% antifreeze is a pure, concentrated solution.
• The 50% solution is half antifreeze and half another substance, likely water.
• The desired 60% solution means that for every part of the solution, 60% is antifreeze.

By creating equations based on these concentrations, we can mathematically determine the exact amounts of each solution to mix.

Linear Equations

Linear equations are mathematical expressions that relate variables in a straight-line format. They often appear in a form like: \[ ax + by = c \]Linear equations are used to solve problems involving relationships between different variables, like the amounts of solutions in this exercise.In Vince's exercise, we made two equations:

• The first is about the total volume: \( x + y = 12 \)
• The second concerns concentration: \( x + 0.5y = 7.2 \)

These equations form a system of linear equations because they are linked by the same variables, \( x \) and \( y \), which represent the quantities of each solution.By solving the system, we can find the precise quantities of each solution needed.

Problem Solving in Algebra

Problem-solving in algebra involves using known strategies and methods to find unknown quantities. In this antifreeze problem, we used a combination of defining variables, setting up equations, and solving them systematically to determine the quantities needed.Here are key steps followed:

• Define Variables: Assigning variables, \( x \) for pure antifreeze, and \( y \) for the 50% solution, helps simplify the problem.
• Set Up Equations: Equations are based on both total quantity and concentration.
• Solve System of Equations: Knowing how to manipulate and solve the equations gives us the answer.
• Verify Solution: Checking back the calculated values in the original conditions ensures correctness.

This approach helps solve many algebraic problems, especially those involving different mixes or blends. It highlights the power of algebra in interpreting and solving real-world situations.