The average (arithmetic mean) length of three pieces of string is 14 inches. Two of the pieces are each 15...

1. TRANSLATE the problem information

• Given information:
  • Average length of three pieces = 14 inches
  • Two pieces are each 15 inches long
  • Need to find the third piece's length

• What this tells us: We can use the average to find the total length of all pieces combined

2. INFER the solution strategy

• Key insight: To find one missing piece when we know the average, we need to:
  • First find the total of all pieces using the average
  • Then subtract what we know to find what's missing

• This is like knowing the total bill at a restaurant was split equally among friends, and finding one person's missing contribution

3. SIMPLIFY to find the total length

• Using the average formula rearranged:
  \(\mathrm{Total\ length = Average × Number\ of\ pieces}\)
  \(\mathrm{= 14 × 3}\)
  \(\mathrm{= 42\ inches}\)

4. SIMPLIFY to find the combined length of known pieces

• Two pieces of 15 inches each: \(\mathrm{15 + 15 = 30\ inches}\)

5. SIMPLIFY to find the third piece

• Length of third piece = Total length - Known pieces
• \(\mathrm{Length\ of\ third\ piece = 42 - 30}\)
  \(\mathrm{= 12\ inches}\)

Answer: B. 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they need to find the total length first. Instead, they might try to work directly with the average and individual pieces, leading to confusion about what operation to perform. They might attempt something like: "If the average is 14 and two pieces are 15, maybe the third piece should also be close to 14?" This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students understand the correct approach but make arithmetic errors. Common mistakes include calculating \(\mathrm{14 × 3 = 32}\) instead of 42, or \(\mathrm{42 - 30 = 10}\) instead of 12. This may lead them to select Choice A (10) instead of the correct answer.

The Bottom Line:

This problem tests whether students can work backwards from an average to find missing information. The key insight is recognizing that the average gives you a direct path to the total, which then makes finding the missing piece straightforward subtraction.