A school district is forming a committee to discuss plans for the construction of a new high school. Of those...

1. TRANSLATE the problem information

• Given information:
  • 15% are parents of students
  • 45% are teachers
  • 25% are school/district administrators
  • 6 remaining individuals are students
• Need to find: How many more teachers than administrators

2. INFER the key insight

• The percentages for parents, teachers, and administrators don't add up to 100%
• This means the 6 students must represent the remaining percentage
• Once I know what percentage 6 students represents, I can work backwards to find the total

3. Find what percentage the students represent

• Parents + Teachers + Administrators = \(\mathrm{15\% + 45\% + 25\% = 85\%}\)
• Students must represent: \(\mathrm{100\% - 85\% = 15\%}\)

4. SIMPLIFY to find the total number of people

• If 6 students = 15% of the total, then:
• \(\mathrm{0.15x = 6}\)
• \(\mathrm{x = 6 \div 0.15 = 40}\) people total

5. Calculate teachers and administrators

• Teachers: \(\mathrm{45\%}\) of 40 = \(\mathrm{0.45 \times 40 = 18}\) people
• Administrators: \(\mathrm{25\%}\) of 40 = \(\mathrm{0.25 \times 40 = 10}\) people

6. Find the difference

• \(\mathrm{18 - 10 = 8}\)

Answer: 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they need to find the total first. Instead, they try to work directly with the given percentages and the number 6, leading to confusion about how to proceed. They might attempt calculations like \(\mathrm{45\% - 25\% = 20\%}\), but then don't know what to do with this percentage since they don't have a total to apply it to. This leads to confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly identify that students represent 15%, but set up the proportion incorrectly. They might write \(\mathrm{\frac{6}{15} = \frac{x}{100}}\) instead of recognizing that 15% of the total equals 6. This incorrect setup leads them to calculate \(\mathrm{6 \times 100 \div 15 = 40}\), which happens to give the right total by coincidence, but shows flawed reasoning that could fail on similar problems.

The Bottom Line:

This problem requires students to work backwards from a 'leftover' group to find the whole, then forward again to find the parts they need. The key insight is recognizing that the 6 students represent the missing piece of the percentage puzzle.