Question Stem:The difference of the squares of two positive integers is 195. If the larger integer is 3 greater than...

1. TRANSLATE the problem information

• Given information:
  • The difference of the squares of two positive integers is 195
  • The larger integer is 3 greater than the smaller integer
  • Need to find: the smaller integer

• Let \(\mathrm{s = smaller\ integer, l = larger\ integer}\)
• This gives us: \(\mathrm{l^2 - s^2 = 195}\) and \(\mathrm{l = s + 3}\)

2. INFER the most efficient approach

• The key insight is recognizing that we have a difference of squares expression
• The difference of squares identity \(\mathrm{a^2 - b^2 = (a + b)(a - b)}\) will help us avoid expanding large expressions
• This transforms our equation into a more manageable form

3. SIMPLIFY using the difference of squares identity

• Apply the identity: \(\mathrm{l^2 - s^2 = (l + s)(l - s)}\)
• Since \(\mathrm{l - s = 3}\), substitute: \(\mathrm{(l + s)(3) = 195}\)
• Divide both sides by 3: \(\mathrm{l + s = 65}\)

4. INFER how to solve the system

• We now have two simple linear equations:
  • \(\mathrm{l - s = 3}\)
  • \(\mathrm{l + s = 65}\)
• Adding these equations eliminates s and gives us l directly

5. SIMPLIFY to find both integers

• Add the equations: \(\mathrm{(l - s) + (l + s) = 3 + 65}\)
• This gives: \(\mathrm{2l = 68}\), so \(\mathrm{l = 34}\)
• Substitute back: \(\mathrm{s = l - 3 = 34 - 3 = 31}\)
• Verify: \(\mathrm{34^2 - 31^2 = 1156 - 961 = 195}\) ✓

Answer: 31

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students might try to expand \(\mathrm{(s + 3)^2 - s^2}\) directly without recognizing the difference of squares pattern, leading to unnecessary complexity. While this approach can work, it's more error-prone and doesn't leverage the elegant structure of the problem. Some students may make arithmetic errors during the expansion or struggle to simplify \(\mathrm{6s + 9 = 195}\) correctly.

Second Most Common Error:

Poor SIMPLIFY execution: Students may correctly set up the system of equations but make errors when adding or subtracting the equations, or make basic arithmetic mistakes when solving for the variables. For instance, they might incorrectly calculate that \(\mathrm{2l = 68}\) means \(\mathrm{l = 33}\), leading to \(\mathrm{s = 30}\), which would give a difference of \(\mathrm{33^2 - 30^2 = 1089 - 900 = 189 \neq 195}\).

The Bottom Line:

This problem rewards students who can recognize structural patterns (difference of squares) and work systematically with linear systems. The key breakthrough is seeing that the difference of squares identity transforms a potentially complex problem into a straightforward system of linear equations.