Question

Solve. At the end of the first year of operation for a business, the business recorded a profit of \(\$ 15,280 .\) Two years later the business recorded a profit of \(\$ 80,450\). What was the percent of increase in profit?

Short answer

Answer: The percent increase in profit is approximately 426.23%.

Step-by-step solution

Find the difference in profit

Subtract the profit of the first year from the profit three years later: Profit_difference = Profit_3_years_later - Profit_first_year = \(80,450 - \)15,280 = $65,170

Calculate the percentage increase

To find the percentage increase in profit, divide the difference in profit by the profit of the first year and multiply by 100: Percentage_increase = (Profit_difference / Profit_first_year) * 100 = (\(65,170 / \)15,280) * 100

Simplify the expression and find the percentage increase

Divide the numbers and multiply by 100 to get the percentage increase: Percentage_increase ≈ (4.2623) * 100 Percentage_increase ≈ 426.23% The percent increase in profit is approximately 426.23%.

Key concepts

Profit Calculation

Understanding profit calculation is essential for any business or financial analysis. Profit is the positive financial gain or benefit realized when the revenue generated from business activities exceeds the expenses, costs, and taxes needed to sustain the activity. Simply put, it's the money a business keeps after paying off all the costs.

Mathematically, profit can be calculated using the formula:
\[ \text{Profit} = \text{Revenue} - \text{Expenses} \]
When analyzing profit over different periods, one might look at how much the profit has increased or decreased. To determine the increase in profit, you can subtract the initial profit from the profit at a later period:
\[ \text{Profit Increase} = \text{Profit at later period} - \text{Profit at earlier period} \]
In our exercise, the business’s profit increase is calculated by subtracting the first year’s profit from the profit recorded two years later. The calculated profit difference is an absolute increase, giving a tangible amount gained over a specified period.

Percentage Increase

The percentage increase is a measure of the relative change in quantity or size compared to the original value, expressed in percent. It's a crucial concept to grasp since it helps to understand growth rates in various contexts, such as profit, population, or investment returns.

To calculate the percentage increase, the following formula is used:
\[ \text{Percentage Increase} = \left(\frac{\text{New value} - \text{Original value}}{\text{Original value}}\right) \times 100 \% \]
This gives a proportionate measure of the increase, providing a clear picture of the growth in relation to the starting value. For our business example, the percentage increase gives us an insight into the business's performance over the years, reflecting how significantly the profit grew.

Arithmetic Operations

Arithmetic operations are the backbone of various mathematical calculations, including those required for profit calculation and percentage increase. They include addition, subtraction, multiplication, and division and are the building blocks for more complex mathematical concepts.

In the given exercise, all these basic operations are applied:

• Subtraction to find the difference in profits between two periods.
• Division to compare the profit increase with the initial profit and to understand the profit increase proportion.
• Multiplication by 100 to convert the proportion into a percentage.

Each of these steps is critical in getting the correct answer, as omitting or incorrectly applying any of these operations would lead to an inaccurate result. Understanding how to perform these operations allows students not just to solve this specific exercise but to handle numerous real-world problems involving financial mathematics.