A two-digit positive integer has the property that the sum of its digits is 9. Additionally, the integer equals 6...

1. TRANSLATE the problem information

• Given information:
  • Two-digit positive integer
  • Sum of its digits is 9
  • The integer equals 6 times its tens digit plus 18
• What this tells us: We need to set up equations using these constraints

2. TRANSLATE into mathematical notation

• Let \(\mathrm{t}\) = tens digit and \(\mathrm{u}\) = units digit
• The two-digit number = \(\mathrm{10t + u}\)
• Constraint equations:
  • \(\mathrm{t + u = 9}\) (sum of digits)
  • \(\mathrm{10t + u = 6t + 18}\) (value constraint)

3. SIMPLIFY the second equation

• Start with: \(\mathrm{10t + u = 6t + 18}\)
• Subtract \(\mathrm{6t}\) from both sides: \(\mathrm{4t + u = 18}\)
• Now we have the system:
  • \(\mathrm{t + u = 9}\)
  • \(\mathrm{4t + u = 18}\)

4. INFER the best solution method and solve

• Since both equations have "+u", elimination is efficient
• Subtract first equation from second: \(\mathrm{(4t + u) - (t + u) = 18 - 9}\)
• This gives: \(\mathrm{3t = 9}\), so \(\mathrm{t = 3}\)
• Substitute back: \(\mathrm{3 + u = 9}\), so \(\mathrm{u = 6}\)

5. APPLY CONSTRAINTS and verify

• The two-digit integer is 36
• Check: \(\mathrm{3 + 6 = 9}\) ✓
• Check: \(\mathrm{6(3) + 18 = 36}\) ✓

Answer: B (36)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misinterpret "6 times its tens digit plus 18" as "6 times (tens digit plus 18)" instead of "(6 times tens digit) plus 18."

This creates the wrong equation: \(\mathrm{10t + u = 6(t + 18) = 6t + 108}\), leading to \(\mathrm{4t + u = 108}\). Combined with \(\mathrm{t + u = 9}\), this gives \(\mathrm{3t = 99}\), so \(\mathrm{t = 33}\), which is impossible for a single digit. Students realize something is wrong but may guess among the answer choices that have digits summing to 9.

This leads to confusion and guessing among Choices A, C, D, or E.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make algebraic errors when solving the system, such as incorrectly subtracting equations or making sign errors.

For example, they might incorrectly get \(\mathrm{t = 4}\) instead of \(\mathrm{t = 3}\), leading them to calculate the number as 45 (since \(\mathrm{4 + 5 = 9}\)). When they check, they find \(\mathrm{6(4) + 18 = 42 \neq 45}\), but they may select it anyway thinking they made a minor error.

This may lead them to select Choice C (45).

The Bottom Line:

This problem tests whether students can accurately translate word constraints into algebra and systematically solve a simple system. The key challenge is parsing the English correctly - many constraint problems hinge on precise interpretation of mathematical relationships described in words.