Two positive numbers have a sum of 18. The product of the two numbers is 80. What is the value...

1. TRANSLATE the problem information

• Given information:
  • Two positive numbers sum to 18
  • Their product is 80
  • Need to find the larger number
• This gives us: \(\mathrm{x + y = 18}\) and \(\mathrm{xy = 80}\)

2. INFER the solution approach

• We have two equations with two unknowns - this is a system
• Since one equation involves a product, substitution will create a quadratic
• Alternative: Look for factor pairs of 80 that sum to 18

3. SIMPLIFY using substitution

• From \(\mathrm{x + y = 18}\), get \(\mathrm{y = 18 - x}\)
• Substitute: \(\mathrm{x(18 - x) = 80}\)
• Expand: \(\mathrm{18x - x^2 = 80}\)
• Rearrange: \(\mathrm{x^2 - 18x + 80 = 0}\)

4. SIMPLIFY the quadratic equation

• Factor: \(\mathrm{(x - 10)(x - 8) = 0}\)
• Solutions: \(\mathrm{x = 10}\) or \(\mathrm{x = 8}\)
• If \(\mathrm{x = 10}\), then \(\mathrm{y = 8}\)
• If \(\mathrm{x = 8}\), then \(\mathrm{y = 10}\)

5. APPLY CONSTRAINTS to select final answer

• Both solutions give us the same pair: 8 and 10
• Since we want the larger number: 10

Answer: C (10)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make errors when expanding \(\mathrm{x(18 - x) = 80}\) or factoring the resulting quadratic.

Common mistakes include:

• Getting signs wrong: writing \(\mathrm{x^2 + 18x + 80 = 0}\) instead of \(\mathrm{x^2 - 18x + 80 = 0}\)
• Factoring incorrectly or being unable to factor \(\mathrm{x^2 - 18x + 80 = 0}\)
• Making arithmetic errors when checking factor pairs

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Inadequate INFER reasoning: Students recognize they need to set up equations but don't see the connection to quadratic equations or factor pairs.

They might try solving \(\mathrm{x + y = 18}\) and \(\mathrm{xy = 80}\) without a systematic approach, leading to trial-and-error that doesn't efficiently explore all possibilities. This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem requires students to bridge word problems with quadratic equations - a connection that many students don't automatically make. Success depends on either systematic algebraic manipulation or organized exploration of factor relationships.