A customer's monthly water bill was $75.74. Due to a rate increase, her monthly bill is now $79.86. To the...

1. TRANSLATE the problem information

• Given information:
  • Original bill: \(\$75.74\)
  • New bill: \(\$79.86\)
  • Need: percent increase to nearest tenth
• The phrase "by what percent did the amount increase" means we need to find percent increase using the formula: \(\frac{\mathrm{new\ value} - \mathrm{original\ value}}{\mathrm{original\ value}} \times 100\%\)

2. INFER the solution approach

• Percent increase problems always compare the change to the original amount
• We need three steps: find the change, divide by original, convert to percent and round

3. SIMPLIFY through the calculations

• Find the absolute increase:
  \(\$79.86 - \$75.74 = \$4.12\)
• Apply the percent increase formula:
  \((\$4.12 \div \$75.74) \times 100\%\)
• Calculate the division (use calculator):
  \(\$4.12 \div \$75.74 = 0.0544...\)
• Convert to percentage:
  \(0.0544... \times 100\% = 5.44...\%\)

4. APPLY CONSTRAINTS for the final answer

• Round to the nearest tenth of a percent:
  \(5.44...\% \approx 5.4\%\)

Answer: D. 5.4%

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Using the new bill amount as the denominator instead of the original amount.

Students might think "the bill is now $79.86, so I should divide by $79.86" and calculate \((\$4.12 \div \$79.86) \times 100\% = 5.16...\% \approx 5.2\%\). This leads them to select Choice C (5.2%).

Second Most Common Error:

Poor TRANSLATE reasoning: Misunderstanding what the problem is asking for and just finding the absolute difference.

Students might calculate \(\$79.86 - \$75.74 = \$4.12\) and think this $4.12 represents "4.1%" without doing the percentage calculation. This causes them to select Choice A (4.1%).

The Bottom Line:

Percent increase problems require careful attention to which value serves as the base for comparison - it's always the original value, never the new value. The key insight is that "percent increase" specifically measures how much the original amount grew.