The number of coins in a collection increased from 9 to 90. What was the percent increase in the number...

1. TRANSLATE the problem information

• Given information:
  • Original number of coins: 9
  • New number of coins: 90
  • Need to find: percent increase

2. INFER the approach

• Percent increase means: "How much did the collection grow as a percentage of its original size?"
• This requires the percent increase formula: \(\frac{\mathrm{new - old}}{\mathrm{old}} \times 100\%\)
• First find the actual increase, then see what percent that represents of the original amount

3. Find the actual increase

• \(\mathrm{Increase = New\,value - Original\,value}\)
• \(\mathrm{Increase = 90 - 9 = 81\,coins}\)

4. SIMPLIFY using the percent increase formula

• \(\mathrm{Percent\,increase} = \frac{\mathrm{increase}}{\mathrm{original}} \times 100\%\)
• \(\mathrm{Percent\,increase} = \frac{81}{9} \times 100\%\)
• \(81 \div 9 = 9\)
• \(9 \times 100\% = 900\%\)

Answer: D. 900%

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about percent increase: Students often think the increase itself (81) represents the percent increase, leading them to select Choice B (81%). They miss that percent increase asks "what percent of the original amount was the increase?"

Second Most Common Error:

Weak INFER skill: Students may calculate that \(90 \div 9 = 10\) and select Choice A (10%), not recognizing this represents "how many times larger" rather than "percent increase." They confuse multiplicative relationships with percent change.

The Bottom Line:

Percent increase problems require understanding that we're comparing the change to the original amount, not just stating the change. The key insight is that a 900% increase means the collection became 10 times its original size \(\mathrm{(original + 900\%\,of\,original = 100\% + 900\% = 1000\% = 10\,times)}\).