Problem 100 - Algebra for College Students

Question

A large corporation has five divisions. Two of the divisions had earnings of \(\$ 2,300,000\) each. The other three divisions had a loss of \(\$ 1,450,000\), a loss of \(\$ 640,000\), and a gain of \(\$ 1,850,000\), respectively. What was the net gain (or loss) of the corporation for the year?

Step-by-Step Solution

Answer

The corporation had a net gain of \( \$ 4,360,000 \).

Step 1: Calculating Total Gains

To determine the total gains of the corporation, we need to sum up the earnings of the divisions that reported gains. The earnings are:- Division 1: \( \\( 2,300,000 \)- Division 2: \( \\) 2,300,000 \)- Division 5: \( \\( 1,850,000 \)Adding these amounts gives:\[ \\) 2,300,000 + \\( 2,300,000 + \\) 1,850,000 = \\( 6,450,000 \]So, the total gains are \( \\) 6,450,000 \).

Step 2: Calculating Total Losses

Next, we need to add up the losses from the divisions that reported losses. This includes:- Division 3: \( \\( 1,450,000 \) loss- Division 4: \( \\) 640,000 \) lossCalculating the total losses:\[ \\( 1,450,000 + \\) 640,000 = \\( 2,090,000 \]So, the total losses are \( \\) 2,090,000 \).

Step 3: Calculating Net Gain or Loss

To find the net gain or loss, subtract the total losses from the total gains:\[ \\( 6,450,000 - \\) 2,090,000 = \\( 4,360,000 \]This result indicates a net gain. Thus, the corporation had a net gain of \( \\) 4,360,000 \).

Key Concepts

Corporate Earnings

Corporate earnings are vital for understanding a corporation's financial health. These earnings represent the profits that divisions or the entire corporation make over a specific time. In a large corporation with multiple divisions, like the example provided, each division's performance can significantly affect the corporation's total financial outcome.

Each division can report either a gain, when they make more money than they spend, or a loss, when they spend more money than they make. To assess the corporation's success, we sum up the gains from profitable divisions and subtract any reported losses. This difference helps stakeholders understand if the company is moving in a positive or negative economic direction.

Understanding corporate earnings is central to strategy and investment decisions. It guides stakeholders, including investors and management, to make informed decisions about resource allocation, expansion, or cost-cutting measures.

Mathematical Problem-Solving

Mathematical problem-solving involves breaking down large problems into manageable steps to find a solution. This systematic approach is essential when dealing with corporate financial data, such as the earnings and losses of multiple divisions within a corporation.

In this scenario, we took a step-by-step approach:

• First, calculate the total gains by adding up all the earnings reported by the profitable divisions.
• Second, tally the total losses by summing the losses from the underperforming divisions.
• Finally, determine the net gain or loss by subtracting the total losses from the total gains.

This method makes complex problems simpler, allowing any underlying patterns or opportunities to become evident. Problem-solving in mathematics encourages logical thinking, enhances analytical skills, and leads to more accurate and reliable results.

Developing strong problem-solving skills is crucial for both academic success and real-world decision-making.

Addition and Subtraction

Addition and subtraction are fundamental arithmetic operations used to handle financial calculations, such as determining a corporation's net gain or loss over a year. These operations help to consolidate financial information for meaningful analysis.

In our example, we used addition to combine the earnings from divisions that had gains. Calculating the sum of multiple numbers helps to derive the total earnings for the corporation. Similarly, we used addition to sum the losses from the divisions that reported losses, providing a clear picture of the company's total financial setbacks.

Subtraction then helped to find the net gain or loss by finding the difference between the total gains and total losses. This straightforward process is key in various real-world scenarios, allowing individuals and corporations to critically evaluate financial situations and take necessary actions based on that evaluation.

Mastering addition and subtraction equips students and professionals with the tools needed to understand and apply arithmetic concepts in daily financial and business decisions.