The product of two consecutive positive integers is 132. What is the value of the smaller integer?

1. TRANSLATE the problem information

• Given information:
  • Two consecutive positive integers
  • Their product equals 132
  • Need to find the smaller integer

• What this tells us: If we call the smaller integer \(\mathrm{n}\), then the integers are \(\mathrm{n}\) and \(\mathrm{n+1}\)

2. TRANSLATE into an equation

• Set up the equation: \(\mathrm{n(n + 1) = 132}\)
• This captures the key relationship: smaller × larger = 132

3. SIMPLIFY to standard quadratic form

• Expand the left side: \(\mathrm{n^2 + n = 132}\)
• Move everything to one side: \(\mathrm{n^2 + n - 132 = 0}\)
• Now we have a quadratic equation in standard form

4. SIMPLIFY using the quadratic formula

• For \(\mathrm{n^2 + n - 132 = 0}\), we have \(\mathrm{a = 1, b = 1, c = -132}\)
• Apply the quadratic formula: \(\mathrm{n = \frac{-1 ± \sqrt{1 + 4×132}}{2}}\)
• Calculate the discriminant: \(\mathrm{1 + 528 = 529}\)
• Take the square root: \(\mathrm{\sqrt{529} = 23}\) (use calculator)
• Complete the formula: \(\mathrm{n = \frac{-1 ± 23}{2}}\)

5. SIMPLIFY to find both solutions

• \(\mathrm{n = \frac{-1 + 23}{2} = \frac{22}{2} = 11}\)
• \(\mathrm{n = \frac{-1 - 23}{2} = \frac{-24}{2} = -12}\)

6. APPLY CONSTRAINTS to select the valid answer

• Since we need positive integers, reject \(\mathrm{n = -12}\)
• The smaller integer is \(\mathrm{n = 11}\)
• The consecutive integers are 11 and 12
• Verification: \(\mathrm{11 × 12 = 132}\) ✓

Answer: B. 11

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to set up the initial equation correctly. They might try to set up something like \(\mathrm{n + (n+1) = 132}\) (thinking about sum instead of product) or get confused about how to represent consecutive integers algebraically.

This leads to confusion and abandoning systematic solution, resulting in guessing among the answer choices.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students solve the quadratic correctly and get both \(\mathrm{n = 11}\) and \(\mathrm{n = -12}\), but then either pick the wrong value or get confused about which one to choose. Some students might even try to use the absolute value and incorrectly select 12 as the "smaller" integer.

This may lead them to select Choice C (12) by incorrectly reasoning that 12 is somehow the answer.

The Bottom Line:

This problem tests whether students can bridge the gap between word problems and quadratic equations. The translation step is crucial - students must recognize that "consecutive integers" means \(\mathrm{n}\) and \(\mathrm{n+1}\), and "product" means multiplication, not addition.