A scientist studying the life cycle of dragonflies counted the number of dragonflies in a certain habitat each day for...

1. TRANSLATE the problem information

• Given information:
  • February 15: 99 dragonflies
  • Percent increase from January 1 to February 15: \(12.50\%\)
  • Need to find: number of dragonflies on January 1

• What this tells us: We're working backwards from a final amount to find the starting amount.

2. INFER the mathematical relationship

• Since we have a percent increase, the relationship is:
  \(\mathrm{Final\;amount} = \mathrm{Initial\;amount} \times (1 + \mathrm{percent\;increase}/100)\)
• We know the final amount (99) and the percent increase \((12.50\%)\), but need the initial amount
• This means we need to solve: \(99 = \mathrm{Initial\;amount} \times (1 + 12.50/100)\)

3. TRANSLATE the relationship into an equation

• Let \(\mathrm{x}\) = number of dragonflies on January 1
• Convert the percentage: \(12.50\% = 12.50/100 = 0.125\)
• Set up the equation: \(99 = \mathrm{x} \times (1 + 0.125)\)
• SIMPLIFY: \(99 = \mathrm{x} \times 1.125\)

4. SIMPLIFY to solve for x

• Divide both sides by 1.125: \(\mathrm{x} = 99 \div 1.125\)
• Calculate: \(\mathrm{x} = 88\) (use calculator if needed)

Answer: A. 88

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the direction of the percent change calculation and try to find \(12.50\%\) of 99, thinking the problem is asking for a direct percentage calculation rather than working backwards from a result.

They calculate: \(99 \times 0.125 = 12.375\), then either subtract this from 99 (getting 86.625) or get confused about what to do next. This leads to confusion and guessing among the available choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly identify they need the percent increase formula but set it up incorrectly as \(99 = \mathrm{x} + (\mathrm{x} \times 0.125)\) instead of \(99 = \mathrm{x} \times 1.125\), leading them to solve \(99 = 1.125\mathrm{x}\) incorrectly.

This algebraic mistake often results in calculation errors that might lead them toward Choice B (87) or cause them to get stuck and guess.

The Bottom Line:

This problem challenges students to work backwards from a final result using the percent increase formula. The key insight is recognizing that when you know the final amount after a percent increase, you divide by \((1 + \mathrm{percent\;increase}/100)\) to find the original amount.