Problem 27 - Prealgebra

Question

Between which two years was there the greatest percent of decrease in the number of CDs sold? \begin{array}{|c|c|} \hline \text { Year } & \begin{array}{c} \text { CDs Sold } \\ \text { (millions) } \end{array} \\ \hline 2003 & 745.9 \\ \hline 2002 & 803.3 \\ \hline 2001 & 881.9 \\ \hline 2000 & 942.5 \\ \hline 1999 & 938.5 \\ \hline \end{array} F \(1999-2000\) G \(2000-2001\) H \(2001-2002\) J \(2002-2003\)

Step-by-Step Solution

Answer

The greatest percent decrease was between 2001 and 2002.

Step 1: Calculate Decrease from 1999 to 2000

The number of CDs sold in 1999 was 938.5 million and in 2000 was 942.5 million. To find the percent decrease, first calculate the change: \( 942.5 - 938.5 = 4 \) million. Since there is no decrease but rather an increase, the percentage decrease is 0%.

Step 2: Calculate Decrease from 2000 to 2001

The number of CDs sold in 2000 was 942.5 million and in 2001 was 881.9 million. Calculate the decrease: \( 942.5 - 881.9 = 60.6 \) million. The percent decrease is calculated by \( \frac{60.6}{942.5} \times 100 \approx 6.43\% \) .

Step 3: Calculate Decrease from 2001 to 2002

The number of CDs sold in 2001 was 881.9 million and in 2002 was 803.3 million. Calculate the decrease: \( 881.9 - 803.3 = 78.6 \) million. The percent decrease is \( \frac{78.6}{881.9} \times 100 \approx 8.91\% \) .

Step 4: Calculate Decrease from 2002 to 2003

The number of CDs sold in 2002 was 803.3 million and in 2003 was 745.9 million. Calculate the decrease: \( 803.3 - 745.9 = 57.4 \) million. The percent decrease is \( \frac{57.4}{803.3} \times 100 \approx 7.14\% \) .

Step 5: Determine the Greatest Percent Decrease

Compare the percent decreases over each interval: 0% (1999-2000), 6.43% (2000-2001), 8.91% (2001-2002), and 7.14% (2002-2003). The greatest percent decrease is 8.91% between 2001 and 2002.

Key Concepts

Understanding Percent Change

Percent change is a way to express how much a quantity increases or decreases in comparison to its original amount. It's commonly used in data analysis to identify trends or shifts.
In this exercise, we are concerned with percent decrease, which measures how much CD sales reduced over specific time periods. Calculating percent change involves two simple steps:

• First, find the difference between the old value and the new value. If we're considering a decrease, subtract the new value from the old value to get a positive number.
• Then, divide the difference by the original value and multiply by 100 to get the percent change.

In formula form, it is expressed as: \[\text{Percent Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\] This concept is vital for determining which year-to-year interval had the biggest decline in CD sales.

Role of Data Analysis in Identifying Trends

Data analysis helps us make sense of numbers and find patterns, especially in business contexts like CD sales. Here, we are analyzing year-over-year statistics to identify when the market faced its steepest decline.
By focusing on the greatest percent decrease in sales, we gain insights into when consumers' music buying habits started to notably shift, perhaps due to digital music services or other market changes.
Collecting and analyzing data such as annual sales figures allow us to spot influential trends and take strategic action based on those observations.
This type of analysis not only applies to sales but to any fluctuating metric, providing crucial information for decision-making processes in companies and organizations.

Navigating Mathematics Problem-Solving

Mathematics problem-solving calls for a clear and logical approach. Here's how you can tackle exercises like this one: First, clearly define the problem and what it's asking for; in this case, identifying the interval with the greatest percent decrease in CD sales. Next, go through each step systematically:

• List out the data you have, including the number of CDs sold each year.
• Calculate the difference for each interval to find decreases.
• Use the percent change formula for accurate results.
• Compare results to find the highest decrease.

The key to math problem-solving is breaking down a complex task into manageable, logical steps and seeing how each step contributes to the final answer.

A Step-by-Step Solution for Clarity

Following a step-by-step approach ensures you don't miss any critical calculations. Let's dig deeper into the provided solution:Start by calculating the change between each year. For instance, from 2000 to 2001, subtract the CDs sold in 2001 from those in 2000, resulting in a `60.6 million` decrease.
Next, apply the percent change formula: \[\text{Percent Decrease} = \frac{60.6}{942.5} \times 100 \approx 6.43\%\]Repeat these calculations for each subsequent interval. Then compare to determine which interval features the highest percent decrease, which was from 2001 to 2002 at `8.91%`.
By following each step methodically, you ensure accuracy and gain a thorough understanding of how to handle similar problems. This step-by-step logical breakdown is essential for mastering any math problem-solving task.