Problem 61 - Intermediate Algebra

Question

How much water should be added to 20 ounces of a \(15 \%\) solution of alcohol to dilute it to a \(10 \%\) alcohol solution? (EQUATION NOT COPY)

Step-by-Step Solution

Answer

Add 10 ounces of water to dilute the solution to 10% alcohol.

Step 1: Understand the Problem

We want to find out how much water should be added to a 20-ounce solution that contains 15% alcohol so that the solution will have only 10% alcohol.

Step 2: Calculate Pure Alcohol in Initial Solution

First, calculate the amount of pure alcohol in the initial solution. Since the solution is 15% alcohol, compute the amount by multiplying the percentage by the total volume:\[ ext{Pure Alcohol} = rac{15}{100} \times 20 = 3 ext{ ounces of alcohol.}\]

Step 3: Set Up the Equation for Dilution

Let the amount of water added be \(x\) ounces. After adding \(x\) ounces of water, the total solution will be \(20 + x\) ounces. The problem indicates the final solution should be 10% alcohol, so:\[rac{3}{20 + x} = rac{10}{100}\]

Step 4: Solve the Equation

Solve for \(x\) by cross-multiplying and simplifying:\[3 = 0.1\times(20 + x)\]which simplifies to:\[3 = 2 + 0.1x\]Subtract 2 from both sides:\[1 = 0.1x\]Then, divide both sides by 0.1:\[x = 10\]Thus, 10 ounces of water must be added.

Step 5: Verify the Final Solution

After adding 10 ounces of water, the new total volume is 30 ounces. Check the alcohol concentration:\[ ext{Alcohol Concentration} = rac{3}{30} = rac{1}{10} = 10 ext{%}\]The solution matches the desired concentration, verifying that the calculation is correct.

Key Concepts

Percent Concentration

Percent concentration is a way of expressing the amount of a particular substance in a solution. Often used in chemistry and medicine, it shows the ratio of the solute (like alcohol) to the total solution in percentage terms. For example, a 15% alcohol solution means there are 15 parts of alcohol for every 100 parts of the solution. It can be calculated using the formula:

• Percent Concentration = \( \frac{{\text{Amount of Solute}}}{{\text{Total Solution Volume}}} \times 100 \)%

This concept helps us understand how strong or weak a solution is based on its contents. In the context of our exercise, identifying the initial percent concentration of alcohol in the solution is essential for determining how much water needs to be added to dilute it.

Mixture Problems

Mixture problems involve combining two or more substances to create a solution with a desired property or concentration. In these problems, it is common to be asked to adjust the proportions of components in a mixture to achieve a specified concentration. Mixture problems often appear in real-life scenarios, such as creating medication solutions, food recipes, or chemical blends. When working with mixture problems:

• Identify the components involved (e.g., alcohol and water in this exercise).
• Understand the initial conditions of the mixture, like the volume and concentration.
• Define the desired outcomes, such as the final concentration post-mix.
• Set up an equation to relate the initial conditions with the desired outcomes.

In our exercise, the challenge was to find out how much water to add to the original alcohol solution to reduce its concentration to 10%. Knowing how to approach mixture problems helps in systematically working through such practical scenarios.

Algebraic Equations

Algebraic equations are mathematical statements that show how two expressions are equal. They are tools for finding unknown values based on known information. In mixture problems, they are used to relate different variables of the solution, like volume and concentration.To tackle mixture problems with algebra:- Identify the unknown quantity, in this case, the amount of water to add ("\(x\)").- Express the total volume of the solution after adding water: \(20 + x\) ounces.- Set up an equation based on the new concentration: \(\frac{3}{20+x} = \frac{10}{100}\).We use algebra to solve for \(x\) by rearranging and simplifying equations:

• Cross-multiply to eliminate fractions.
• Reorganize the equation to isolate the variable.
• Perform arithmetic operations to solve for the unknown.

In our case, solving the equation helped us find that 10 ounces of water must be added to dilute the alcohol solution to 10% concentration. Mastering algebraic equations can be incredibly helpful in understanding and solving many dilution and mixture-based problems.