Problem 51 - Intermediate Algebra

Question

A pound of dried pineapple bits sells for \(\$ 6.19\), a pound of dried banana chips sells for \(\$ 4.19,\) and a pound of raisins sells for \(\$ 2.39\) a pound. Two pounds of raisins are to be mixed with equal amounts of pineapple and banana to create a trail mix that will sell for \(\$ 4.19\) a pound. How many pounds of pineapple and banana chips should be used?

Step-by-Step Solution

Answer

1.80 pounds of pineapple and 1.80 pounds of banana chips are needed.

Step 1: Define Variables

Let the amount of dried pineapple be \( x \) pounds and the amount of dried banana chips be \( x \) pounds. Since there are equal amounts of both in the mix.

Step 2: Calculate Total Pounds of Trail Mix

Since there are 2 pounds of raisins already, the total pounds of trail mix can be expressed as \( x + x + 2 = 2x + 2 \) pounds.

Step 3: Express Total Cost of Trail Mix

The total cost is calculated by considering the cost of each ingredient: \( 6.19x \) for pineapple, \( 4.19x \) for banana, and \( 2.39 \times 2 = 4.78 \) for raisins.

Step 4: Set Up the Cost Equation

The total cost should equal \( 4.19 \) times the total weight of the trail mix: \( 6.19x + 4.19x + 4.78 = 4.19(2x + 2) \).

Step 5: Simplify and Solve the Equation

Combine like terms in the equation: \( 10.38x + 4.78 = 8.38x + 8.38 \). Subtract \( 8.38x \) from both sides to isolate \( x \) on one side: \( 2x + 4.78 = 8.38 \).

Step 6: Solve for x

Subtract 4.78 from both sides: \( 2x = 3.60 \). Then divide by 2: \( x = 1.80 \).

Step 7: Conclusion

Each of pineapple and banana chips in the trail mix should weigh 1.80 pounds.

Key Concepts

Understanding Cost Equations

Cost equations are essential for solving mixture word problems in algebra. They help us relate the cost of individual components in a mixture to the overall cost of the mixture itself.
In our exercise, each ingredient in the trail mix has a unique price per pound:

• Pineapple costs \(6.19\) dollars per pound.
• Banana chips cost \(4.19\) dollars per pound.
• Raisins cost \(2.39\) dollars per pound.

The idea is to express the total cost of the trail mix using these individual costs, ensuring it matches the desired selling price of \(4.19\) dollars per pound multiplied by the entire mixture's weight.
This process ensures that our mixture is economically feasible, reflecting both the desired price and quality components.

Defining Variables

Variable definition is a critical step in solving algebraic problems. It helps us break down the problem into solvable components by representing unknown quantities with symbols or letters.
In this exercise, we introduce the variable \(x\) as the weight of both the dried pineapple and the dried banana chips, since the problem states that their amounts are equal.
Using the same variable for both components simplifies our equations, allowing us to focus on just one part of the unknown quantity. By setting the weight of both pineapple and banana chips as \(x\) pounds, we can articulate the problem's constraints more clearly and set the stage for constructing meaningful equations.

A Step by Step Solution

Following a step-by-step solution offers a structured way to tackle complex problems. Each step builds logically on the last to finally arrive at a solution.
1. **Define Variables**: Assign \(x\) to the amount of pineapple and banana chips.
2. **Total Weight of Mix**: Calculate by adding the given component weights: \(2x + 2\) pounds total, considering the 2 pounds of raisins included.
3. **Summing Costs**: Reflects both the individual and total ingredient costs, ensuring consistency with desired pricing.4. **Set Up Cost Equation**: Combine everything to form \(6.19x + 4.19x + 4.78 = 4.19(2x + 2)\).
5. **Solve**: Simplify to find \(x\) through basic algebraic manipulations, ultimately solving the problem.Step-by-step guidance minimizes errors and outlines a clear path to the solution.

Solving Equations

Solving equations involves isolating the variable to find its value. It is a fundamental skill in algebra.
In this exercise, once the equation is established:

• Combine like terms: Simplify the expression \(6.19x + 4.19x\) into \(10.38x\).
• Isolate \(x\) by moving terms: Move all terms involving \(x\) to one side of the equation and constants to the other.
• Simplify: Subtract constexpr from both sides and simplify: \(2x + 4.78 = 8.38\) becomes \(2x = 3.60\).
• Divide to find \(x\): Divide \(3.60\) by 2, yielding \(x = 1.80\).

These steps distill complex problems into manageable steps, fostering deeper understanding and efficient problem-solving.