A list contains four consecutive multiples of 3 in increasing order, where the least number in the list is n....

1. TRANSLATE the problem information

• Given information:
  • Four consecutive multiples of 3: \(\mathrm{n, n + 3, n + 6, n + 9}\)
  • Need to find inequality representing the relationship

2. TRANSLATE each part of the key statement

• 'Three times the third number':
  • Third number is \(\mathrm{n + 6}\)
  • Three times this: \(\mathrm{3(n + 6)}\)

• 'Sum of the first and fourth numbers':
  • First number: \(\mathrm{n}\)
  • Fourth number: \(\mathrm{n + 9}\)
  • Sum: \(\mathrm{n + (n + 9)}\)

3. INFER the meaning of 'at least 10 greater than'

• 'At least' means \(\geq\) (greater than or equal to)
• '10 greater than the sum' means (sum + 10)
• So we need: \(\mathrm{3(n + 6) \geq [sum\ of\ first\ and\ fourth] + 10}\)

4. TRANSLATE the complete inequality

• Putting it all together:
  \(\mathrm{3(n + 6) \geq n + (n + 9) + 10}\)

• This matches choice (A) exactly.

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret 'at least 10 greater than' and reverse the inequality direction.

They might think 'at least' means the expression should be less than or equal to something, leading them to write: \(\mathrm{3(n + 6) \leq n + (n + 9) + 10}\)

This may lead them to select Choice B.

Second Most Common Error:

Poor TRANSLATE reasoning: Students mix up which number is which (first vs. third vs. fourth) or misread the structure entirely.

They might identify the wrong number as 'third' or confuse the sum, potentially leading to: \(\mathrm{3(n + 9) \geq n + (n + 6) + 10}\)

This may lead them to select Choice C.

The Bottom Line:

This problem requires careful translation of English to mathematics. The phrase 'at least 10 greater than' is the critical piece - students must recognize this means the left side is greater than or equal to the right side plus 10, not the other way around.