Problem 26 - College Algebra and Calculus: An Applied Approach

Question

The profit for a company in February was \(5 \%\) lower than it was in January. The total profit for the two months was \(\$ 129,000\). Find the profit for each month.

Step-by-Step Solution

Answer

The profit for January was approximately $66,153.85 and for February, it was about $62,846.15.

Step 1: Define Variables

Let's denote the January profit as \(X\). The February profit will consequently be \(X - 0.05X = 0.95X\).

Step 2: Setup the Equation

According to the problem, the total of these two amounts equals $129,000, which leads us to the equation \(X + 0.95X = 129000\).

Step 3: Solve the Equation

Sum the two terms on the left side of the equation to consolidate it to \(1.95X = 129000\).

Step 4: Find the January profit

Solving for \(X\), Jan profit would be \(X = 129000 / 1.95 = 66153.85\).

Step 5: Calculate the February profit

Subtract the percentage from the January profit to get February profit as \(0.95X = 0.95*66153.85 = 62846.15 \).

Key Concepts

Profit Calculation

Profit calculation is an essential part of understanding a business's financial health. When calculating profits, it’s vital to distinguish between different time periods, as profit can vary month to month due to multiple factors.

In the exercise, profits for two months are compared. The first step is to define each month's profit correctly. For January, the profit is labeled as a variable we can work with, "\(X\)". February’s profit is comparatively lower by 5%, which we represent as "\(0.95X\)" - this is because February’s profit is 95% of January’s profit:

• January's profit: \(X\)
• February's profit: \(0.95X\)

This approach makes it straightforward to calculate the total profit from both months, which equals \(\$129,000\). Breaking down the profits into equations allows you to analyze and project financial trends.

Seasonal Comparison

Seasonal comparisons are about analyzing how performance metrics, like profits in this example, change over different times of the year. Businesses often experience fluctuations in performance due to seasonality.

The exercise presents a scenario where February's profits are 5% less than January's. This decrease might be due to seasonal variation like post-holiday slumps. By comparing two consecutive months, you gain insights into how external factors affect business profits:

• Acknowledge factors like demand variation.
• Assess changes like promotional impacts or external economic factors.
• Plan contingencies for times of reduced profit.

These comparisons help businesses strategize for periods where profits might dip, enabling them to plan budgets and resources more efficiently and meet their financial goals. It also aids in understanding patterns over time.

Equation Solving

Equation solving is a fundamental algebra skill useful for resolving real-world problems such as determining profit amounts. In this exercise, we use equations to determine the unknown monthly profits, by translating the given verbal information into mathematical expressions.

1. **Set up the Equation**: Start with the profits as defined in variables, January as \(X\) and February as \(0.95X\). The equation becomes: \[X + 0.95X = 129000\] 2. **Combine Like Terms**: Simplify the left side of the equation by adding the coefficients of \(X\): \[1.95X = 129000\] 3. **Solve for X**: Isolate \(X\) by dividing both sides by 1.95: \[X = \frac{129000}{1.95} = 66153.85\]4. **Find February Profit**: Use the calculated January profit to find February's by multiplying it by 0.95: \[0.95 \times 66153.85 = 62846.15\] Understanding these steps in equation solving allows you to find unknowns based on known conditions efficiently, providing clarity and precision when assessing financial metrics.