Problem 50 - Intermediate Algebra

Question

A store sells regular green tea for \(\$ 16\) a pound and an exotic loose leaf tea for \(\$ 28\) a pound. To get rid of 40 pounds of the exotic loose leaf tea that are not selling, a shopkeeper makes a blend to put on sale for \(\$ 20\) a pound. How many pounds of green tea should he use?

Step-by-Step Solution

Answer

The shopkeeper should use 80 pounds of green tea.

Step 1: Define Variables

Let \( x \) represent the number of pounds of green tea to be used in the blend. Since we are blending with 40 pounds of exotic tea, the total weight of the blend will be \( x + 40 \) pounds.

Step 2: Set Up the Equation for the Blend Cost

The cost of the green tea is \( 16x \) dollars and the cost of the exotic tea is \( 28 \times 40 \) dollars. The total cost of the blend should equal the sale price times the total weight, which is \( 20(x + 40) \). This gives us the equation: \[ 16x + 28 \times 40 = 20(x + 40) \].

Step 3: Simplify the Equation

First calculate \( 28 \times 40 \), which is 1120. Substitute this back into the equation to get: \[ 16x + 1120 = 20(x + 40) \]. Now simplify the right side \( 20(x + 40) \) to \( 20x + 800 \).

Step 4: Solve for x

Bring like terms to the same side: \[ 16x + 1120 = 20x + 800 \]. Subtract \( 16x \) from both sides to get \( 1120 = 4x + 800 \). Then, subtract 800 from both sides: \( 320 = 4x \). Divide by 4 to solve for \( x \): \( x = 80 \).

Step 5: Interpret the Solution

The solution \( x = 80 \) indicates that the shopkeeper should blend 80 pounds of green tea with the 40 pounds of exotic tea to make a blend that can be sold at \( \$20 \) per pound.

Key Concepts

Understanding Linear Equations in Mixture Problems

Linear equations are essentially math sentences that express relationships between variables. They're a pivotal part of algebra and show how changes in one quantity can influence another. In mixture problems like our tea blend situation, linear equations help us figure out how different parts of the mix come together.
Let's break down what happens in the tea problem:

• We need to find out how many pounds of green tea to add to exotic tea.
• A linear equation helps relate the costs and weights of these teas to meet a specific price target.

We establish our equation by considering the total cost of separate components, in this case, green tea and exotic tea, and set it equal to the target cost for the resulting blend. Here we use variables (like \(x\) for the weight of green tea) to represent unknowns and solve for these to complete our number puzzle.
This application of linear equations reveals how seemingly complex money and weight calculations can be neatly resolved through algebra.

Cost Analysis in Creating a Tea Blend

Cost analysis involves examining component expenses to achieve a desired price or profit. This is very applicable in business-related algebra problems. In our example, the shopkeeper wants a blend priced at \\(20 per pound. Let's see how cost analysis applies here:

• The green tea costs \\)16 per pound.
• The exotic tea, originally \$28 per pound, seems unsellable unless mixed.

For the blend:

• Calculate the initial cost of 40 pounds of exotic tea: \(28 \times 40 = 1120\).
• The cost for mixing with \(x\) pounds of green tea equals \(16x\).
• The total cost needs to match \(20(x + 40)\) given the combined resale price.

This analysis ensures that despite high starting costs, the final blend remains financially viable. Cost analysis aids in making business decisions that balance inventory management and pricing strategies.

Algebraic Expressions and Their Role

Algebraic expressions form the bedrock of algebra and they play a critical role in solving mixture problems like ours. They consist of numbers, letters, and arithmetic operations, and algebra uses them to translate real-world problems into math language. Here's how they're used in this exercise:

• The blend includes two ingredients with their own costs, modeled through expressions.
• The green tea's cost is \(16x\), where \(x\) is its weight.
• The exotic tea's cost is a fixed \(28 \times 40\), or \(1120\) dollars.

Putting these together gives us the equation: \[16x + 28 \times 40 = 20(x + 40)\]Solving this cleverly crafted expression lets us discover the optimal amount of green tea needed. This power of algebraic expressions helps break down the task into manageable steps, making it easier to handle even complex-looking problems with confidence.