Problem 259 - Elementary Algebra

Question

A scientist needs 65 liters of a \(15 \%\) alcohol solution. She has available a \(25 \%\) and a \(12 \%\) solution. How many liters of the \(25 \%\) and how many liters of the \(12 \%\) solutions should she mix to make the \(15 \%\) solution?

Step-by-Step Solution

Answer

Mix 15 liters of 25% solution with 50 liters of 12% solution.

Step 1: Define Variables

Let x be the number of liters of the 25% solution. Let y be the number of liters of the 12% solution.

Step 2: Set Up the Equations

Use the information given to set up two equations. The first equation represents the total volume of the mixtures: \[ x + y = 65 \]. The second equation represents the total amount of alcohol in the mixtures: \[ 0.25x + 0.12y = 0.15 \times 65 \].

Step 3: Simplify the Second Equation

Simplify the second equation by calculating the right-hand side: \[ 0.15 \times 65 = 9.75 \]. Hence, the second equation becomes: \[ 0.25x + 0.12y = 9.75 \].

Step 4: Solve the System of Equations

Solve the system of equations:From the first equation: \[ y = 65 - x \] Substitute for y in the second equation: \[ 0.25x + 0.12(65 - x) = 9.75 \] This simplifies to: \[ 0.25x + 7.8 - 0.12x = 9.75 \] \[ 0.13x + 7.8 = 9.75 \] \[ 0.13x = 1.95 \] \[ x = 15 \]

Step 5: Find y

Substitute \( x = 15 \) back into the first equation to find y: \[ y = 65 - 15 = 50 \].

Step 6: Conclusion

Therefore, to make 65 liters of a 15% alcohol solution, mix 15 liters of the 25% solution with 50 liters of the 12% solution.

Key Concepts

System of Equations

In algebra, a system of equations is a collection of two or more equations with the same set of variables. In this exercise, the scientist is working with two unknowns: the amount of 25% alcohol solution (x) and the amount of 12% alcohol solution (y). We create two equations based on the information provided. The first equation represents the total volume: The second equation reflects the concentration of alcohol in the mixture: To resolve, we need to solve this system simultaneously, often by substitution or elimination methods.

Percentage Concentration

Percentage concentration is a way of expressing the amount of a substance in a solution as a proportion of the total volume. Here, we are dealing with alcohol percentages in two solutions: 25% and 12%. The goal is to obtain a 15% solution from the mixture. In fractional terms, a 25% solution has 0.25 liters of alcohol per liter of solution, and a 12% solution has 0.12 liters of alcohol per liter of solution. Balancing these percentages in the final mix ensures the desired concentration.

Algebraic Substitution

Algebraic substitution is a technique used to solve systems of equations. After setting up our system, we solve one equation for one variable and then substitute this expression into the other equation. Here, from the equation Next, we substitute this expression for y into the second equation: By substituting for y, we simplify the problem to a single equation with one variable, which we can solve straightforwardly.

Solving Linear Equations

Solving linear equations involves manipulating equations to find the values of the variables. Using our substitution, we end up with: We simplify to: Now, solve for x: Finally, we substitute back to find y, using our first equation: This systematic process demonstrates the importance of each step: setting up equations, substituting correctly, and solving diligently. In this exercise, we verified that the scientist needs 15 liters of the 25% solution and 50 liters of the 12% solution to achieve her desired mixture.