Problem 25 - Elementary Algebra

Question

Thirty ounces of a punch that contains \(10 \%\) grapefruit juice is added to 50 ounces of a punch that contains \(20 \%\) grapefruit juice. Find the percent of grapefruit juice in the resulting mixture.

Step-by-Step Solution

Answer

The resulting punch mixture contains 16.25% grapefruit juice.

Step 1: Define the Variables

Let's define the parts of the problem: We have two different punches and we want to find the percent of grapefruit juice in the resulting mixture. Let \( x_1 \) be the amount of grapefruit juice in the first punch and \( x_2 \) be the amount in the second punch.

Step 2: Calculate the Amount of Grapefruit Juice in Each Punch

For the first punch, which has 30 ounces and \(10\%\) grapefruit juice, the amount of juice, \( x_1 \), is calculated as: \[ x_1 = 0.10 imes 30 = 3 \text{ ounces} \]For the second punch, which has 50 ounces and a \(20\%\) grapefruit juice content, \( x_2 \) is: \[ x_2 = 0.20 imes 50 = 10 \text{ ounces} \]

Step 3: Find the Total Amount of Grapefruit Juice in the Mixture

Add \( x_1 \) and \( x_2 \) to find the total amount of grapefruit juice in the mixture: \[ x_{ ext{total}} = x_1 + x_2 = 3 + 10 = 13 \text{ ounces} \]

Step 4: Calculate the Total Volume of the Mixture

Sum the volumes of the two punches to find the total volume of the resulting mixture:\[ V_{ ext{total}} = 30 + 50 = 80 \text{ ounces} \]

Step 5: Determine the Percent of Grapefruit Juice in the Mixture

The percent of grapefruit juice in the mixture is the total ounces of grapefruit juice divided by the total ounces of the mixture: \[ \text{Percent of grapefruit juice} = \left( \frac{x_{\text{total}}}{V_{\text{total}}} \right) \times 100 = \left( \frac{13}{80} \right) \times 100 = 16.25\%\]

Step 6: Conclusion

The resulting mixture has a \(16.25\%\) grapefruit juice content.

Key Concepts

Understanding Mixture Problems

Mixture problems are a common type of algebraic exercise that involves combining two or more substances to form a new mixture. These problems often require us to find a component's concentration or percentage in the resulting mixture. To solve a mixture problem, it is crucial to understand the properties of each component involved, including their quantities and concentration levels.
In our grapefruit juice problem, we are dealing with two different punch mixtures. One has a concentration of 10% grapefruit juice in 30 ounces, while the other contains 20% grapefruit juice in 50 ounces. The objective is to determine the percentage of grapefruit juice in the final mixture after combining these two punches.
Solving mixture problems effectively often involves setting up equations that represent the relationship between the quantities and concentrations. By doing this, we can simplify the problem into manageable steps that will lead to the desired solution.

Mastering Percent Calculations

Percent calculations are vital in solving mixture problems, as they are used to express the concentration of a component within a mixture. Understanding how to calculate percentages helps in solving these types of problems accurately.
To find the percentage of a substance in a mixture, we first need to determine the amount of the substance in each portion of the mix. For instance, in the grapefruit juice example, we calculate the amount of grapefruit juice by taking 10% of 30 ounces for the first punch and 20% of 50 ounces for the second punch:

• For the first punch: \( x_1 = 0.10 \times 30 = 3 \) ounces
• For the second punch: \( x_2 = 0.20 \times 50 = 10 \) ounces

Adding these gives us the total grapefruit juice, which can then be used to find the percentage of the juice in the final mixture. This step-by-step method ensures clarity and precision when handling percent calculations.
Once we have the total grapefruit juice, we divide it by the total volume of the mixture, multiplying by 100 to convert it to a percentage. Practicing these types of calculations regularly helps solidify the understanding of percentages and their role in mixture problems.

Importance of Step-by-Step Problem Solving

Step-by-step problem solving is essential for tackling complex algebraic problems, such as mixture problems. It involves breaking down the problem into smaller, more manageable parts and solving each step sequentially.
In our example, we first defined the variables to clearly understand the task at hand. Then, we calculated the amount of grapefruit juice in each punch and summed them to find the total juice content. Next, we determined the total volume of the resulting mixture, which allowed us to compute the final percentage of grapefruit juice.
Having a methodical approach provides several advantages:

• It reduces errors by ensuring each step is completed before moving to the next.
• It provides a clear record of the solution process, which is valuable for reviewing or explaining the work.
• It builds problem-solving skills, as it encourages logical and structured thinking.

By consistently applying step-by-step problem solving, students can improve their ability to tackle algebraic problems with confidence and accuracy.