From 2005 through 2014, the number of music CDs sold in the United States declined each year by approximately 15%...

1. TRANSLATE the problem information

• Given information:
  • CD sales decline by 15% each year
  • Initial sales in 2005: 600 million CDs
  • t = years after 2005
  • Need to find model for C (millions of CDs sold)

2. INFER what "decline by 15%" means mathematically

• If sales decline by 15% each year, then each year retains:
  \(\mathrm{100\% - 15\% = 85\%}\) of the previous year
• This means we multiply by 0.85 each year
• This describes exponential decay with retention rate 0.85

3. INFER the exponential model structure

• Exponential decay follows: \(\mathrm{C = (initial\ value) \times (retention\ rate)^t}\)
• Initial value = 600 million
• Retention rate = 0.85
• Time variable = t
• Therefore: \(\mathrm{C = 600(0.85)^t}\)

4. APPLY CONSTRAINTS to eliminate incorrect choices

• For exponential decay, the base must be between 0 and 1
• Choice A: 0.15 is between 0 and 1 ✓, but represents what's lost, not what remains
• Choice B: 0.85 is between 0 and 1 ✓ and represents what remains
• Choices C & D: 1.15 and 1.85 are > 1, indicating growth not decay ✗

Answer: B. \(\mathrm{C = 600(0.85)^t}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students focus on the "15%" and mistakenly use 0.15 as the base instead of understanding that declining by 15% means retaining 85%.

Their reasoning: "It says 15% decline, so the base should be 0.15"

This may lead them to select Choice A (\(\mathrm{C = 600(0.15)^t}\))

Second Most Common Error:

Conceptual confusion about decay vs. growth: Students mix up the direction of change and think "15% decline" should be modeled as "15% added to 100%" giving 115% = 1.15.

Their reasoning: "Decline by 15% means we use 1.15 somehow"

This may lead them to select Choice C (\(\mathrm{C = 600(1.15)^t}\))

The Bottom Line:

The key insight is recognizing that exponential decay requires understanding what percentage remains after each decline, not what percentage is lost. Declining by 15% means keeping 85%, so the base is 0.85, not 0.15.