Jeremy deposited x dollars in his investment account on January 1, 2001. The amount of money in the account doubled...

1. TRANSLATE the problem information

• Given information:
  • Initial deposit: \(\mathrm{x}\) dollars (January 1, 2001)
  • Money doubles each year
  • Final amount: $480 (January 1, 2005)
  • Find: \(\mathrm{x}\)

2. INFER the time period and growth pattern

• From January 1, 2001 to January 1, 2005 = 4 years
• "Doubles each year" means we multiply by 2 each year
• This creates exponential growth: \(\mathrm{x × 2 × 2 × 2 × 2 = x × 2^4}\)

3. TRANSLATE into an equation

• After 4 years: \(\mathrm{x × 2^4 = 480}\)

4. SIMPLIFY the equation

• Calculate the exponent: \(\mathrm{2^4 = 16}\)
• So we have: \(\mathrm{x × 16 = 480}\)
• Solve for x: \(\mathrm{x = 480 ÷ 16 = 30}\)

Answer: 30

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students often miscalculate the time period, thinking there are 5 years from 2001 to 2005 instead of 4 years.

If they use 5 years, they would set up: \(\mathrm{x × 2^5 = 480}\), giving \(\mathrm{x × 32 = 480}\), so \(\mathrm{x = 15}\). This fundamental misunderstanding of the time period leads to an incorrect answer.

Second Most Common Error:

Poor SIMPLIFY execution: Students may correctly identify the exponential relationship but make errors calculating \(\mathrm{2^4}\), perhaps thinking \(\mathrm{2^4 = 8}\) instead of 16.

This would lead them to solve \(\mathrm{x × 8 = 480}\), giving \(\mathrm{x = 60}\), which is double the correct answer. Simple computational errors in exponent evaluation can significantly impact the final result.

The Bottom Line:

This problem requires careful attention to time intervals and systematic application of exponential growth. Students must resist the temptation to simply count calendar years and instead focus on the number of growth periods that occur.