Problem 36 - Algebra A Combined Function

Question

A local lumberyard uses quantity pricing. The table shows the price per board for different amounts of lumber purchased. $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Price per Board } \\ \text { (in dollars) } \end{array} & \begin{array}{c} \text { Number of Boards } \\ \text { Purchased } \end{array} \\ \hline 8.00 & 1 \\ \hline 7.50 & 10 \\ \hline 6.50 & 25 \\ \hline 5.00 & 50 \\ \hline 2.00 & 100 \\ \hline \end{array} $$ a. Write the data as ordered pairs of the form (price per board, number of boards purchased). b. In your own words, write the meaning of the ordered pair (2.00,100) . c. Create a scatter diagram of the paired data. Be sure to label the axes appropriately. d. What trend in the paired data does the scatter diagram show?

Step-by-Step Solution

Answer

The price per board decreases as the number of boards purchased increases.

Step 1: Create Ordered Pairs

To write the data as ordered pairs of the form (price per board, number of boards purchased), simply pair the price per board with the corresponding number of boards. The ordered pairs are: (8.00, 1), (7.50, 10), (6.50, 25), (5.00, 50), and (2.00, 100).

Step 2: Interpret the Ordered Pair

The ordered pair (2.00, 100) means that when 100 boards are purchased, the price per board is $2.00. This shows a discount example; buying in bulk significantly decreases the cost per unit.

Step 3: Draw the Scatter Diagram

Create a scatter diagram using the ordered pairs. Label the x-axis as 'Number of Boards Purchased' and the y-axis as 'Price per Board'. Plot the points: (1, 8.00), (10, 7.50), (25, 6.50), (50, 5.00), and (100, 2.00) on this graph.

Step 4: Analyze the Scatter Diagram

After plotting the points on the scatter diagram, you will notice a downward trend. This indicates that as the number of boards purchased increases, the price per board decreases.

Key Concepts

Quantity Pricing

Quantity pricing is a pricing strategy where the price of an item varies depending on the quantity purchased. This is often implemented to encourage customers to buy more. The more items you buy, the cheaper the price per item usually becomes.
This approach is widely used in various industries, such as retail and manufacturing.

• It provides an incentive for bulk purchases by offering discounts on larger quantities.
• Such discounts can be a win-win situation, as businesses increase their volume sales while customers benefit from lower prices.

In this exercise, we observe quantity pricing through board prices, where buying more boards results in a lower cost per board.

Ordered Pairs

An ordered pair, like (price per board, number of boards purchased), gives us a straightforward representation of two related quantities. In our lumberyard exercise, each ordered pair represents a specific price per board associated with a certain quantity of boards.
For example, the ordered pair (8.00, 1) signifies that if you buy 1 board, each costs $8.00.

• Order matters in ordered pairs. The first element is the price, the second is the quantity.
• This concept helps organize data and easily understand relationships between two variables.

Understanding ordered pairs is central in plotting data on a graph for graphical representation and analysis.

Trend Analysis

Trend analysis involves examining patterns in data to predict future movements. The scatter plot created by plotting the ordered pairs provides visual insight into these patterns.
In the context of our exercise, as you plot the ordered pairs in a scatter diagram, you notice the pattern's direction.

• A downward trend appears, indicating as more boards are bought, the price per board decreases.
• Such analysis is crucial for businesses to understand how pricing strategies, like quantity pricing, affect sales and customer behavior.

Observing trends can help businesses make data-driven decisions and highlight the effectiveness of their pricing strategies.

Bulk Pricing Discount

Bulk pricing discounts are a practical application of quantity pricing, where larger orders lead to lower prices per unit.
This is evident from our data: buying 100 boards reduces the price per board to $2.00, compared to $8.00 for a single board.
Such discounts are beneficial in encouraging larger purchases.

• Consumers are encouraged to buy in larger quantities to gain cost savings, leading to increased sales volume for sellers.
• These discounts can help businesses clear large inventory or meet sales targets efficiently.

Bulk pricing is a strategic approach to pricing that leverages volume to attract customers while optimizing sales and inventory management.