Problem 29 - Understanding Elementary Algebra with Geometry

Question

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. A coffee wholesaler wishes to produce 60 pounds of a coffee blend selling at \(\$ 3.10\) per pound. How many pounds of coffee blends selling at \(\$ 3.35\) per pound and \(\$ 2.75\) per pound should be mixed to produce such a mixture?

Step-by-Step Solution

Answer

35 pounds at \$3.35 per pound and 25 pounds at \$2.75 per pound.

Step 1: Define the Variables

Let \( x \) be the pounds of coffee selling at \( \$3.35 \) per pound and \( y \) be the pounds of coffee selling at \( \$2.75 \) per pound.

Step 2: Set Up the Equations

We need to form two equations based on the total weight and the total cost. The first equation will represent the total weight: \[ x + y = 60 \] The second equation will represent the total cost: \[ 3.35x + 2.75y = 3.10 \times 60 \]

Step 3: Simplify the Cost Equation

Calculate the right-hand side of the second equation: \[ 3.35x + 2.75y = 186 \]

Step 4: Solve the System of Equations - Substitute

From the first equation, isolate \( y \): \[ y = 60 - x \] Substitute \( y = 60 - x \) into the second equation: \[ 3.35x + 2.75(60 - x) = 186 \]

Step 5: Solve for \( x \)

Distribute and combine like terms: \[ 3.35x + 165 - 2.75x = 186 \] \[ 0.60x + 165 = 186 \] Subtract 165 from both sides: \[ 0.60x = 21 \] \[ x = 35 \]

Step 6: Solve for \( y \)

Now substitute \( x = 35 \) back into the first equation: \[ y = 60 - 35 \] \[ y = 25 \]

Step 7: Verify the Solution

Verify by checking if the total cost is correct: \[ 3.35 \times 35 + 2.75 \times 25 = 3.10 \times 60 \] \[ 117.25 + 68.75 = 186 \] \[ 186 = 186 \] The solution is correct.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that express the equality between two expressions. In this exercise, we defined two algebraic equations based on the problem's conditions. The first equation was about the total weight of the coffee blend, and the second one was based on the total cost. Here are the steps:

• Identify variables to represent unknown quantities, like the pounds of different coffee blends.
• Create equations that represent the relationships described in the problem.
• Solve these equations simultaneously to find the unknown values.

By carefully translating the problem's wordy descriptions into algebraic expressions, we can use mathematical tools to find a precise solution.

Verbal Problems

Verbal problems, or word problems, present a scenario that requires translation into mathematical terms. In our exercise, we needed to figure out how much of each coffee blend to mix to achieve a specific price and weight. Here’s how to approach verbal problems:

• Read the problem carefully and identify key information and requirements.
• Define variables for unknown quantities.
• Create equations based on the relationships and conditions given in the problem.

Breaking down the problem into smaller parts and systematically converting the text into mathematical statements helps solve even the trickiest verbal problems.

Mixture Problems

Mixture problems involve combining different substances to achieve a specific mixture with desired properties. In our coffee blend example, the goal was to mix two types of coffee to achieve a blend selling at a specific price per pound. Here's a general approach:

• Define variables for each component in the mixture.
• Create an equation for the total amount of the mixture.
• Form another equation for the overall cost or value of the mixture.

Solving these equations together provides the amounts of each component needed. Effective handling of mixture problems relies on a clear understanding of the relationships among quantities and values involved.

Elementary Algebra

Elementary algebra deals with the basics of algebra, including understanding variables, forming and solving equations, and manipulating expressions. This exercise used basic algebra concepts, such as:

• Isolating variables to solve for unknowns in equations.
• Substituting values to find solutions sequentially.
• Verifying solutions by checking them against the original problem conditions.

By mastering elementary algebra, students can confidently tackle more complex problems and build strong mathematical foundations. This exercise helps illustrate how simple algebraic techniques apply to practical situations like calculating mixture proportions and costs.