Problem 5 - Thinking Mathematically

Question

In Exercises 5-8, one more piece of information is given than is necessary for solving the problem. Identify this unnecessary piece of information. Then solve the problem. A salesperson receives a weekly salary of \(\$ 350\). In addition, \(\$ 15\) is paid for every item sold in excess of 200 items. How much extra is received from the sale of 212 items?

Step-by-Step Solution

Answer

The extra amount received from the sale of 212 items is $180.

Step 1: Identify the excess items sold

The excess items sold are calculated as the total items sold minus 200. So, the excess items sold = 212 - 200 = 12.

Step 2: Calculate the extra amount received

The extra amount received is found by multiplying the number of excess items by the amount received for each excess item sold. So, the extra amount received = 12 items * $15/item = $180.

Key Concepts

Problem-Solving

In order to address any problem effectively, it's important to adopt a structured approach. First, carefully read the problem to comprehend the facts and the question being asked. In our example, we are provided with a salesperson's salary parameters and need to determine extra income based on sales. By accurately identifying key components such as base salary, item threshold, and additional payment conditions, the solution becomes clearer.
Break down the problem into manageable parts and focus on the steps provided. This involves recognizing the critical data necessary to solve the problem while disregarding any extraneous information that might cause confusion. Utilizing these strategies not only aids in solving mathematical problems but also enhances overall analytical skills in varied scenarios.

Mathematical Reasoning

Mathematical reasoning is pivotal in solving problems and justifying solutions. It involves logical thinking and processing information in a structured manner. In this exercise, we apply mathematical reasoning to deduce the number of excess items beyond a certain constraint, which is 200 items in this case.
Begin by determining known values, such as total sales and the set threshold. Subtraction helps find excess values. Then, apply multiplication to quantify the outcome, establishing how much those extra sales translate into financial gain. That calculation reveals the additional earning based on the excess sales. Strategy and clarity in reasoning allow you to accurately perform these calculations without error.

Excess Items Calculation

Calculating excess items involves identifying how many units exceed a particular limit. In this exercise, the focus is on sales exceeding 200 items. Here's how to approach it:

• Start with the total number of items sold, which is 212.
• Subtract the threshold of 200 items from the total to find excess items.
• The calculation would be: 212 - 200 = 12 excess items.

This simple subtraction tells us how many items surpass the base amount. It’s crucial because it sets the stage for further calculations regarding bonus or extra payment. By mastering the concept of excess calculation, you gain tools that can be applied to various real-world scenarios, such as managing inventory or setting sales targets.

Income Calculation

Income calculation, particularly in commission-based structures, involves understanding how additional earnings are computed. Here, the salesperson earns extra income through sales above a certain threshold.
For this particular situation:

• Each excess item sells for an extra $15.
• With 12 excess items, the extra income is calculated by multiplying: 12 items x $15 per item = $180.

This result shows the additional earnings apart from the base salary, providing an insight into incentive-driven income models. By practicing these calculations, you're better equipped to assess financial outcomes and understand compensation plans in professional settings.