Question

Sporting Goods Sales In 2002, the total sales of sporting goods in the United States was \(\$ 77,726,000,000\). In 2003 , the total sales increased \(2.6 \%\) from 2002 . In 2004 , the total sales increased \(6.1 \%\) from 2003 . In 2005, the total sales increased \(2.5 \%\) from 2004. (Source: National Sporting Goods Association) (a) Find the total sporting goods sales in 2003 . (b) Find the total sporting goods sales in 2004 . (c) Find the total sporting goods sales in 2005 . (d) Find the percent increase in total sales from 2002 to \(2005 .\)

Short answer

(a) The total sporting goods sales in 2003 was the outcome of Step 1. (b) The total sporting goods sales in 2004 was the outcome of Step 2. (c) The total sales for 2005 came from Step 3. (d) The percent increase in total sales from 2002 to 2005 was the result of Step 4.

Step-by-step solution

Calculate Total Sales for 2003

To calculate the total sales for 2003, add the percentage increase, 2.6%, to the total sales of 2002. To apply a percentage increase, multiply the original amount, $77,726,000,000, by \(1 + \frac{2.6}{100} = 1.026\). Thus, the total sales for 2003 is \(\$77,726,000,000 \times 1.026\).

Calculate Total Sales for 2004

To calculate the total sales for 2004, add the percentage increase of 6.1% to total sales in 2003, found in the previous step. Calculate it as the result of Step 1 multiplied by \(1 + \frac{6.1}{100} = 1.061\).

Calculate Total Sales for 2005

Sales in 2005 increased 2.5% from the 2004's sales. Apply the same methodology: multiply the 2004 total sales by \(1 + \frac{2.5}{100} = 1.025\).

Calculate Percent Increase from 2002 to 2005

To calculate the total percent increase, subtract the total sales from 2002 from the total sales of 2005, divide that result by the total sales of 2002, and then multiply by 100. Mathematically, \[\% \text{{Increase}} = \left( \frac{{\text{{Sales}}_{2005} - \text{{Sales}}_{2002}}}{{\text{{Sales}}_{2002}}} \right) \times 100\].

Key concepts

Algebraic Expressions

Algebraic expressions are a cornerstone of mathematics, enabling us to represent relationships between variables and constants symbolically. In the context of percentage increase calculations, they allow us to quantify changes over time or across different scenarios. For example, when we talk about the 2.6% increase in total sales for 2003, we use the algebraic expression \( 77,726,000,000 \times (1 + \frac{2.6}{100}) \), where \( \frac{2.6}{100} \) represents the percentage change converted to a decimal.

Understanding how to construct and manipulate these expressions is crucial for solving problems in applied mathematics and beyond. When dealing with percentage increases, an original value (often represented as a variable or constant) is multiplied by \(1 + \frac{\text{percent change}}{100}\) to find the new value after the increase. The expression encapsulates the initial value, the rate of increase, and the resulting final value in a concise mathematical statement.

Percent Change

Percent change, a measure of relative change expressed as a percentage, is used to compare the difference between two values. As seen in our sporting goods sales example, percent change helps us understand the growth or decline over time. Calculating percent change involves a few steps: First, find the difference between the new value and the original value. Then, divide this difference by the original value. Finally, multiply the result by 100 to get a percentage.

For instance, to calculate the 2.6% increase for 2003, we start with the 2002 sales and apply the percentage increase formula \( \text{New Value} = \text{Original Value} \times (1 + \frac{\text{Percent Change}}{100}) \). In practice, this looks like \( \text{New Value} = 77,726,000,000 \times (1 + \frac{2.6}{100}) \), which simplifies our calculations and offers a straightforward method for finding subsequent years' sales, organizing data, and making predictions.

Applied Mathematics

Applied mathematics involves using mathematical methods to solve real-world problems. Percentage increase calculations are a prime example of applied mathematics, as they allow businesses and economists to analyze trends and make informed decisions. In our textbook exercise, understanding how sporting goods sales have grown over specific years requires both arithmetic and algebraic skills. Applied mathematics takes abstract concepts and puts them into practice; in this case, we take the abstract idea of 'percentage growth' and apply it to actual sales figures to model and understand economic trends.

When we speak of applied mathematics, we are not only looking at the methods but also the interpretation of results. After calculating the sales figures for 2003, 2004, and 2005, as well as the overall percent increase from 2002 to 2005, we have a numerical representation of the business's performance. These figures can be used to inform strategies, prepare forecasts, and set goals for future growth – demonstrating the real-world significance and power of mathematical concepts and calculations.