United States Presidents from 1789 to 2015AgesNumber40-44245-49750-541355-591160-64765-693The table above gives the number of United States presidents...

1. TRANSLATE the problem information

• Given information:
  • Table showing age ranges and number of presidents in each range
  • Need fraction of presidents "at least 50" who were "at least 60"

• What this tells us:
  • "At least 50" means ages 50 and above (includes 50-54, 55-59, 60-64, 65-69)
  • "At least 60" means ages 60 and above (includes 60-64, 65-69)
  • We're looking for a conditional fraction: (presidents \(\mathrm{≥60\ AND\ ≥50}\)) ÷ (presidents \(\mathrm{≥50}\))

2. INFER the approach

• This is a conditional probability situation - we need to restrict our sample space
• The denominator should be all presidents who were at least 50 years old
• The numerator should be those who were at least 60 years old (who are automatically also at least 50)

3. Calculate the denominator (total presidents at least 50 years old)

• Add up the age groups 50 and above:
  • Ages 50-54: 13 presidents
  • Ages 55-59: 11 presidents
  • Ages 60-64: 7 presidents
  • Ages 65-69: 3 presidents
  • Total: \(\mathrm{13 + 11 + 7 + 3 = 34}\) presidents

4. Calculate the numerator (presidents at least 60 years old)

• Add up the age groups 60 and above:
  • Ages 60-64: 7 presidents
  • Ages 65-69: 3 presidents
  • Total: \(\mathrm{7 + 3 = 10}\) presidents

5. Form the fraction

• Fraction = (Presidents at least 60) ÷ (Presidents at least 50)
• Fraction = \(\mathrm{\frac{10}{34}}\)

Answer: B. \(\mathrm{\frac{10}{34}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret the sample space and use all presidents (43 total) as the denominator instead of restricting to those at least 50 years old.

They calculate: All presidents = \(\mathrm{2 + 7 + 13 + 11 + 7 + 3 = 43}\), then form the fraction \(\mathrm{\frac{10}{43}}\).

This leads them to select Choice A (\(\mathrm{\frac{10}{43}}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand "at least 50" and only include the 50-59 age groups in their sample space, excluding the 60+ groups.

They calculate: Presidents 50-59 = \(\mathrm{13 + 11 = 24}\), then form the fraction \(\mathrm{\frac{10}{24}}\) (still correctly identifying 10 as the numerator).

This leads them to select Choice C (\(\mathrm{\frac{10}{24}}\)).

The Bottom Line:

This problem tests your ability to work with conditional situations where you need to restrict your sample space. The key insight is recognizing that "of those presidents who were at least 50" creates a new, smaller universe to work within, not the entire set of all presidents.