Problem 67 - College Algebra

Question

For the following exercises, use this scenario: A health-conscious company decides to make a trail mix out of almonds, dried cranberries, and chocolate- covered cashews. The nutritional information for these items is shown in Table 1 . $$ \begin{array}{|c|c|c|c|} \hline & \text { Fat (g) } & \text { Protein (g) } & \text { Carbohydrates (g) } \\ \hline \text { Almonds (10) } & 6 & 2 & 3 \\ \hline \text { Cranberries (10) } & 0.02 & 0 & 8 \\ \hline \text { Cashews (10) } & 7 & 3.5 & 5.5 \\ \hline \end{array} $$ For the "hiking" mix, there are 1,000 pieces in the mix, containing \(390.8 \mathrm{~g}\) of fat, and \(165 \mathrm{~g}\) of protein. If there is the same amount of almonds as cashews, how many of each item is in the trail mix?

Step-by-Step Solution

Answer

The mix contains 300 almonds, 400 cranberries, and 300 cashews.

Step 1: Define Variables for Quantities

Let \( x \) be the number of almonds, \( y \) the number of dried cranberries, and \( z \) the number of cashews in the mix. From the problem, we know \( x = z \). Therefore, we only need to deal with two variables \( x \) and \( y \).

Step 2: Set Up Initial Equations

We are told that there are 1,000 pieces in the mix. Thus, \( x + y + z = 1000 \). Since \( x = z \), this equation becomes \( 2x + y = 1000 \).

Step 3: Equation for Fat Content

The fat contribution from the almonds is \( 0.6x \), from the cranberries is \( 0.002y \), and from the cashews is \( 0.7z \) which equals \( 0.7x \). The total fat is 390.8 g. Therefore, \( 0.6x + 0.002y + 0.7x = 390.8 \). Simplifying this, we have \( 1.3x + 0.002y = 390.8 \).

Step 4: Equation for Protein Content

For protein, the almonds contribute \( 0.2x \), the cranberries contribute \( 0y \), and the cashews contribute \( 0.35z = 0.35x \). The total protein is 165 g, giving \( 0.2x + 0.35x = 165 \). Simplifying, we have \( 0.55x = 165 \).

Step 5: Solve for \( x \)

From \( 0.55x = 165 \), solve for \( x \) by dividing both sides by 0.55: \[ x = \frac{165}{0.55} = 300 \].

Step 6: Solve for \( y \)

Substitute \( x = 300 \) back into the equation from Step 2: \( 2(300) + y = 1000 \). So, \( 600 + y = 1000 \). Solving for \( y \), we get \( y = 400 \).

Step 7: Confirm with Additional Condition

Finally, as \( x = z \), we also have \( z = 300 \). Using all these values, check the initial conditions: \( 2x + y = 1000 \), \( 1.3x + 0.002y = 390.8 \), and \( 0.55x = 165 \) all hold true. Therefore, the solution satisfies all conditions.

Key Concepts

Systems of Equations

In algebra, systems of equations are used to find the values of variables that satisfy multiple equations at the same time. This involves finding a common solution set that works for all equations. In our trail mix problem, we used systems of equations to balance the number of items, their fat content, and protein content. Each element in this mix contributes differently to the total nutritional value, and our task was to find how many pieces of each item there are. To start, we defined our variables: \( x \) for almonds, \( y \) for cranberries, and \( z \) for cashews. Given that almonds and cashews are equal (\( x = z \)), we simplified our equations. This means we had fewer variables to solve for compared to constraints, which generally simplifies solving systems of equations. The first equation \( 2x + y = 1000 \) comes from the total number of pieces, whereas the equations for fat \( 1.3x + 0.002y = 390.8 \) and protein \( 0.55x = 165 \) come from their respective contents. Solving these systems effectively pinpoints the exact number of each type of item in the mix.

Fat and Protein Calculation

Fat and protein content calculations are crucial for understanding nutritional values, especially in food production or when creating a balanced diet. In this problem, we combined the fat and protein contributions from each ingredient to match the overall totals.

• **Almonds:** Contribute \( 0.6 \) grams of fat and \( 0.2 \) grams of protein per 10 pieces.
• **Cranberries:** Have almost no fat with just \( 0.002 \) grams but have no protein.
• **Cashews:** Contribute \( 0.7 \) grams of fat and \( 0.35 \) grams of protein per 10 pieces, similar in contribution to the almonds.

By summing up the contributions from each type, we set up equations to represent the total fat \( 1.3x + 0.002y = 390.8 \) and total protein \( 0.55x = 165 \). This allows us to find a proportionate distribution of these components in the trail mix, ensuring it matches the specified nutritional content.

Quantitative Reasoning

Quantitative reasoning involves using mathematical skills to solve real-world problems. It requires understanding, interpreting, and analyzing numerical data to make decisions. This trail mix problem is an excellent exercise in quantitative reasoning as it combines logical thinking with mathematical calculations. In this scenario, we had to interpret given nutritional data and translate it into mathematical equations. The definitions of the equations from simple variable expressions to the final solutions showcase how algebra can model real-life situations. Using a strategy to initially define product similarities—such as almonds equating to cashews—simplifies complex problems. It’s often strategic to exploit given symmetries or equalities to reduce what initially seems complicated into a more manageable form. Practicing these skills enhances our ability to tackle similar problems, making quantitative reasoning a valuable asset in both academic and everyday contexts.