The sum of two positive numbers is 84. Their difference is 26. What is the value of the larger number?

1. TRANSLATE the problem information

• Given information:
  • Two positive numbers have a sum of 84
  • The same two numbers have a difference of 26
  • Need to find the larger number

• What this tells us:
  If \(\mathrm{L = larger\ number}\) and \(\mathrm{S = smaller\ number}\), then:
  • \(\mathrm{L + S = 84}\)
  • \(\mathrm{L - S = 26}\)

2. INFER the solution strategy

• We have two equations with two unknowns - this is a system of equations
• Adding the equations will eliminate S and give us L directly
• This is more efficient than substitution for this particular problem

3. SIMPLIFY by adding the equations

• \(\mathrm{(L + S) + (L - S) = 84 + 26}\)
• \(\mathrm{L + S + L - S = 110}\)
• \(\mathrm{2L = 110}\)
• \(\mathrm{L = 55}\)

Answer: C. 55

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often mix up which number is larger when setting up equations. They might write \(\mathrm{S - L = 26}\) instead of \(\mathrm{L - S = 26}\), not recognizing that "difference" means larger minus smaller in this context.

This leads them to get \(\mathrm{L = 29}\) instead of \(\mathrm{L = 55}\), causing them to select Choice A (29).

Second Most Common Error:

Poor INFER reasoning: Students recognize they need a system but choose substitution over elimination. While this works, it's more complex and creates opportunities for algebraic mistakes. Some students get confused during the substitution process and abandon the systematic approach.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can efficiently translate a word problem into a clean system of equations. The key insight is recognizing that adding equations eliminates one variable immediately, making this a straightforward one-step solve rather than a complex algebraic manipulation.