The sum of a positive integer x and its reciprocal is 10/3. What is the value of x?

1. TRANSLATE the problem information

• Given information:
  • A positive integer x plus its reciprocal equals 10/3
  • Need to find the value of x

• Mathematical equation: \(\mathrm{x + \frac{1}{x} = \frac{10}{3}}\)

2. SIMPLIFY by eliminating the fraction

• Multiply both sides by x to clear the reciprocal:

\(\mathrm{x^2 + 1 = \frac{10}{3}x}\)

• Multiply everything by 3 to eliminate the remaining fraction:

\(\mathrm{3x^2 + 3 = 10x}\)

• Rearrange to standard quadratic form:

\(\mathrm{3x^2 - 10x + 3 = 0}\)

3. SIMPLIFY by factoring the quadratic

• Factor \(\mathrm{3x^2 - 10x + 3 = 0}\)
• Look for two numbers that multiply to \(\mathrm{(3)(3) = 9}\) and add to -10
• Those numbers are -9 and -1
• Factor: \(\mathrm{(3x - 1)(x - 3) = 0}\)

4. SIMPLIFY to find both solutions

• Set each factor equal to zero:

\(\mathrm{3x - 1 = 0}\) → \(\mathrm{x = \frac{1}{3}}\)

\(\mathrm{x - 3 = 0}\) → \(\mathrm{x = 3}\)

5. APPLY CONSTRAINTS to select the final answer

• The problem specifies x must be a positive integer
• 1/3 is positive but not an integer
• 3 is both positive and an integer
• Therefore: \(\mathrm{x = 3}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS reasoning: Students solve the quadratic correctly but fail to recognize that 1/3 is not an integer. They see 1/3 as an answer choice and select it without checking the constraint.

This may lead them to select Choice A (1/3).

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when clearing fractions or factoring the quadratic equation. Common mistakes include sign errors when rearranging or incorrect factoring patterns.

This leads to confusion and incorrect solutions that don't match any answer choice, causing them to guess.

The Bottom Line:

This problem tests whether students can work systematically through equation-solving while remembering to check constraints. The key insight is that mathematical solutions must still satisfy the original problem conditions - not every algebraic solution is a valid real-world answer.