Problem 262 - Elementary Algebra

Question

A \(90 \%\) antifreeze solution is to be mixed with a \(75 \%\) antifreeze solution to get 360 liters of a \(85 \%\) solution. How many liters of the \(90 \%\) and how many liters of the \(75 \%\) solutions will be used?

Step-by-Step Solution

Answer

Use 240 liters of 90% antifreeze and 120 liters of 75% antifreeze.

Step 1: Define Variables

Let \( x \) be the volume of the 90% antifreeze solution and \( y \) be the volume of the 75% antifreeze solution.

Step 2: Establish Volume Equation

We need 360 liters in total, so the first equation is: \[ x + y = 360 \]

Step 3: Establish Antifreeze Concentration Equation

The mixture must result in an 85% antifreeze solution. Therefore, the second equation based on the percentage of antifreeze is: \[ 0.90x + 0.75y = 0.85 \times 360 \]

Step 4: Simplify the Second Equation

Solve the right side first: \[ 0.85 \times 360 = 306 \]. Thus, the second equation becomes: \[ 0.90x + 0.75y = 306 \]

Step 5: Solve the System of Equations

From the volume equation, express \( y \): \[ y = 360 - x \]. Substitute this into the second equation: \[ 0.90x + 0.75(360 - x) = 306 \]

Step 6: Solve for \( x \)

Expand and combine like terms: \[ 0.90x + 270 - 0.75x = 306 \]. This simplifies to: \[ 0.15x = 36 \]. Solve for \( x \): \[ x = 240 \]

Step 7: Solve for \( y \)

Substitute \( x = 240 \) back into \( y = 360 - x \): \[ y = 360 - 240 = 120 \]

Key Concepts

System of Equations

When solving mixture problems in algebra, we often deal with a system of equations. This means we have multiple equations that share common variables. For our problem, we set up two equations:

• One for the total volume of the mixture.
• One for the concentration of the mixture.

By solving these equations together, we find the values of the variables that satisfy both. In our exercise, we used the volume equation ( x + y = 360 ) and the concentration equation ( 0.90x + 0.75y = 306 ). We can solve such systems using substitution or elimination methods. Both methods aim to simplify the equations step by step until we find the values of all variables involved.

Percentage Concentration

Percentage concentration indicates the amount of a substance in a mixture expressed as a percentage of the total volume. In our problem:

• The 90% antifreeze solution means 90% of its volume is antifreeze.
• The 75% solution means 75% of its volume is antifreeze.
• We aim for an 85% solution in the final mixture.

We convert these percentages into decimals when forming our equations for easier calculations. For instance, 90% becomes 0.90, and 75% becomes 0.75. Thus, when setting up our concentration equation, we multiplied these decimals by the respective volumes of each solution.

Volume Calculation

To tackle mixture problems, we often need to calculate total volumes by combining different solutions. Here, we labeled the volume of the 90% solution as x and the volume of the 75% solution as y . The resulting volume of the mixture was given as 360 liters.
An important step was creating an equation to represent this combined volume:

• x + y = 360

This equation ensures the sum of the individual volumes equals the total volume we desire. By solving for x and y , we determined the quantities of each solution needed to achieve our target volume.

Linear Equations

Linear equations are equations of the first degree, meaning they involve no exponents or powers higher than one. In our solution, both the volume and concentration equations were linear:
x + y = 360 and 0.90x + 0.75y = 306 . These linear equations formed the system we solved.
Linear equations can be easily graphed as straight lines. The solution to our system of equations corresponds to the point where both lines intersect. For algebraic solutions, we manipulated these linear equations through substitution or elimination to find the variable values. This process ensures that we meet the problem's conditions perfectly.