The table summarizes the number of objects in each group.GroupNumber of objectsA375B54C690D81Total1,200The number of objects in group C is p%...

1. TRANSLATE the problem information

• Given information:
  • Group A has 375 objects
  • Group C has 690 objects
  • Group C is \(\mathrm{p\%}\) of Group A

• What this tells us: We need to find what percentage 690 is of 375

2. TRANSLATE the percentage relationship into an equation

• The phrase "690 is p% of 375" becomes: \(\mathrm{690 = \frac{p}{100} \times 375}\)
• This is the fundamental percentage equation: \(\mathrm{Part = \frac{Percent}{100} \times Whole}\)

3. SIMPLIFY the equation to solve for p

• First, multiply out the right side:
  \(\mathrm{690 = \frac{p}{100} \times 375 = 3.75p}\)
• Now divide both sides by 3.75:
  \(\mathrm{p = 690 \div 3.75}\)
• Calculate the division (use calculator):
  \(\mathrm{p = 184}\)

Answer: 184

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse which number should be the "whole" and which should be the "part"

They might incorrectly think: "375 is p% of 690" and set up \(\mathrm{375 = \frac{p}{100} \times 690}\), leading to \(\mathrm{p = 375 \div 6.9 \approx 54.3}\). This leads to confusion since 54.3 isn't a clean answer, causing them to abandon systematic solution and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students make calculation errors when computing \(\mathrm{690 \div 3.75}\)

They might approximate incorrectly (like \(\mathrm{690 \div 4 = 172.5}\)) or make division mistakes, leading to answers that don't match the expected result. This causes them to second-guess their setup and potentially guess randomly.

The Bottom Line:

This problem tests whether students can correctly translate a percentage statement into mathematical form and then execute the arithmetic accurately. The key insight is recognizing that "C is p% of A" means \(\mathrm{C = \frac{p}{100} \times A}\), not the reverse.