Two positive integers have a product of 240. The difference between the larger integer and the smaller integer is 8....

1. TRANSLATE the problem information

• Given information:
  • Two positive integers have product 240
  • The difference between larger and smaller is 8
  • Need to find the smaller integer

• Setting up variables: Let \(\mathrm{x}\) = smaller integer, \(\mathrm{y}\) = larger integer

2. TRANSLATE constraints into equations

• From the given information:
  • \(\mathrm{xy = 240}\) (product condition)
  • \(\mathrm{y - x = 8}\) (difference condition)

3. INFER the solution strategy

• We have two equations with two unknowns - this suggests using substitution
• From the simpler equation: \(\mathrm{y = x + 8}\)
• Substitute this into the product equation to get one equation in one variable

4. SIMPLIFY through substitution and expansion

• Substitute \(\mathrm{y = x + 8}\) into \(\mathrm{xy = 240}\):
  \(\mathrm{x(x + 8) = 240}\)

• Expand and rearrange:
  \(\mathrm{x^2 + 8x = 240}\)
  \(\mathrm{x^2 + 8x - 240 = 0}\)

5. SIMPLIFY by factoring the quadratic

• Need two numbers that multiply to -240 and add to 8
• Testing systematically: \(\mathrm{20 \times (-12) = -240}\) and \(\mathrm{20 + (-12) = 8}\) ✓
• Factor: \(\mathrm{(x + 20)(x - 12) = 0}\)
• Solutions: \(\mathrm{x = -20}\) or \(\mathrm{x = 12}\)

6. APPLY CONSTRAINTS to select the valid solution

• Since we need positive integers: \(\mathrm{x = 12}\)
• Therefore: \(\mathrm{y = 12 + 8 = 20}\)
• Verify: \(\mathrm{12 \times 20 = 240}\) ✓ and \(\mathrm{20 - 12 = 8}\) ✓

Answer: 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may try to guess and check with factors of 240 instead of setting up the systematic equation approach. They might test pairs like (1,240), (2,120), (3,80), etc., looking for a difference of 8, but this becomes tedious and they may miss the correct pair (12,20) or make arithmetic errors in the process.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly solve the quadratic to get \(\mathrm{x = -20}\) or \(\mathrm{x = 12}\), but fail to recognize that \(\mathrm{x}\) must be positive. They might select the negative value or become confused about which solution to choose.

This may lead them to incorrectly consider -20 as a valid answer or to make an arbitrary choice between the two solutions.

The Bottom Line:

This problem requires students to bridge word problems with quadratic equations - a connection many students don't naturally make. The key insight is recognizing that substitution transforms this into a standard quadratic factoring problem.