Problem 73 - Understanding Elementary Algebra with Geometry

Question

Solve each of the following problems algebraically. Be sure to label what the variable represents. A beaker contains \(40 \mathrm{ml}\) of a \(60 \%\) alcohol solution. What percentage of alcohol solution must be added to produce \(70 \mathrm{ml}\) of a \(45 \%\) solution?

Step-by-Step Solution

Answer

25% alcohol solution must be added.

Step 1 Title - Define the variables

Let the percentage of the alcohol solution that must be added be represented by the variable x. This represents the percentage of the alcohol in the solution that will be combined with the existing solution.

Step 2 Title - Set up the equation

Create an equation based on the alcohol content. The initial amount of pure alcohol is 40 ml * 60% = 24 ml. The final amount of solution needed is 70 ml * 45% = 31.5 ml. Let V be the volume of the added solution. Then the equation is: 24 ml + x * V = 31.5 ml

Step 3 Title - Solve for the added solution volume

The total volume of the final solution is 70 ml. Thus, the added solution volume V is: 70 ml - 40 ml = 30 ml

Step 4 Title - Substitute volume and solve for x

Substitute V = 30 ml into the equation 24 + x * 30 = 31.5 Solve for x: 24 + 30x = 31.5 30x = 31.5 - 24 30x = 7.5 x = 7.5 / 30 x = 0.25 or 25%

Key Concepts

Variable Definition

Understanding the problem starts with defining the unknown value, which is represented by a variable. In this case, the variable is **x**, which symbolizes the percentage of the new alcohol solution we need to add. Defining variables helps in organizing the information and setting a clear path to solving the problem.

Here’s how you do it:

• Identify what you're solving for: the percentage of the new alcohol solution.
• Let the variable **x** represent this unknown percentage.

By setting **x** as the percentage of alcohol to be added to the beaker, we establish what we are looking for in the equation.

Equation Setup

The next step is to set up an equation that links all the given data. For our problem, we’ll focus on the quantity of alcohol in both the initial and final solutions. This step takes our variable and incorporates it into a mathematical structure.

Follow these steps:

• Calculate the initial amount of pure alcohol: 40 ml * 60% = 24 ml.
• Determine the amount of alcohol needed in the final solution: 70 ml * 45% = 31.5 ml.
• Express the final total amount as: 24 ml (initial) + **x** * volume of added solution = 31.5 ml (final).

So, the setup equation will be: 24 ml + **x** * V = 31.5 ml, where **V** is the volume of the added solution.

Solution Volume Calculation

To move forward with solving the equation, it’s important to know how much new solution we are adding. This step fills in another key piece of the puzzle.

Here’s what we do next:

• Recognize that the final volume of the solution is given: 70 ml.
• The beaker started with 40 ml.
• So, the added solution's volume is: V = 70 ml - 40 ml = 30 ml.

Thus, we determine that **V**, the volume of the added solution, is 30 ml.

Substitution and Solving Equations

With the volumes calculated and the equation set up, we can substitute the known values and solve for our variable **x**. This is the core of algebra: simplifying equations to find unknowns.

Here’s how to proceed:

• Start with the substitution: 24 + **x** * 30 = 31.5.
• Simplify the equation: 30x = 31.5 - 24.
• Calculate: 30x = 7.5.
• Find **x**: x = 7.5 / 30 = 0.25 or 25%.

This means that the added solution needs to be 25% alcohol to achieve the desired 45% alcohol solution in the final 70 ml mixture.