A father is currently three times as old as his son. In 6 years, the father will be twice as...

1. TRANSLATE the problem information

• Given information:
  • Father is currently 3 times as old as son
  • In 6 years, father will be 2 times as old as son will be then
  • Need to find father's current age
• What this tells us: We have two different time periods (now and 6 years from now) with different age relationships

2. INFER the approach

• This problem involves relationships at two different times, so we need a system of equations
• Let \(\mathrm{s}\) = son's current age and \(\mathrm{f}\) = father's current age
• We'll set up one equation for each time period, then solve

3. TRANSLATE each relationship into equations

• Current relationship: \(\mathrm{f = 3s}\)
• Future relationship (6 years from now): \(\mathrm{f + 6 = 2(s + 6)}\)
  • Father's age in 6 years = \(\mathrm{f + 6}\)
  • Son's age in 6 years = \(\mathrm{s + 6}\)
  • Father will be twice the son's age then

4. SIMPLIFY by substituting and solving

• Substitute \(\mathrm{f = 3s}\) into the second equation:
  \(\mathrm{3s + 6 = 2(s + 6)}\)
• Expand:
  \(\mathrm{3s + 6 = 2s + 12}\)
• Collect like terms:
  \(\mathrm{3s - 2s = 12 - 6}\)
• Solve:
  \(\mathrm{s = 6}\)

5. Find father's current age

• Since \(\mathrm{f = 3s}\) and \(\mathrm{s = 6}\):
  \(\mathrm{f = 3(6) = 18}\)

Answer: C. 18

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often struggle to correctly set up the future age relationship equation. They might write \(\mathrm{f + 6 = 2s}\) instead of \(\mathrm{f + 6 = 2(s + 6)}\), forgetting that the son will also be 6 years older.

This incorrect equation leads to:
\(\mathrm{3s + 6 = 2s}\) → \(\mathrm{s = -6}\) (impossible age)

This creates confusion and typically leads to guessing among the answer choices.

Second Most Common Error:

Poor INFER reasoning: Students might try to solve this with just one equation, not recognizing that they need both the current and future relationships to create a solvable system.

They might only use \(\mathrm{f = 3s}\) and get stuck because they have two unknowns but only one equation. This leads to abandoning systematic solution and guessing.

The Bottom Line:

Age relationship problems require careful attention to time periods. The key insight is recognizing that "in 6 years" affects BOTH people's ages, not just one person's age. Students must translate each time period separately and then connect them through the same variables.