A software company charges $180 for a license that covers up to 3 users. For each additional user beyond the...

1. TRANSLATE the problem information

• Given information:
  • Base license cost: $180 for up to 3 users
  • Additional user cost: $45 for each user beyond the first 3
  • Need function C(n) for n users where \(\mathrm{n \geq 3}\)

2. INFER the cost structure

• This is a two-part pricing model: base cost + additional charges
• Key insight: For n users total, the number of "additional users beyond the first 3" is \(\mathrm{(n - 3)}\)
• Total cost = Base cost + (Additional users) × (Cost per additional user)

3. TRANSLATE into mathematical expression

• \(\mathrm{C(n) = 180 + (n - 3) \times 45}\)
• \(\mathrm{C(n) = 180 + 45(n - 3)}\)

4. SIMPLIFY the expression

• \(\mathrm{C(n) = 180 + 45(n - 3)}\)
• \(\mathrm{C(n) = 180 + 45n - 135}\)
• \(\mathrm{C(n) = 45n + 45}\)

5. Verify with test cases

• For \(\mathrm{n = 3}\): \(\mathrm{C(3) = 45(3) + 45 = 180}\) ✓
• For \(\mathrm{n = 4}\): \(\mathrm{C(4) = 45(4) + 45 = 225}\) (should be $180 + $45 = $225) ✓

Answer: (A) \(\mathrm{C(n) = 45n + 45}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students misunderstand what "additional users beyond the first 3" means and think there are n additional users instead of \(\mathrm{(n - 3)}\) additional users.

They set up the function as \(\mathrm{C(n) = 180 + 45n}\), reasoning that you pay $180 base plus $45 for each of the n users. This leads them to select Choice (B) (\(\mathrm{C(n) = 45n + 180}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{180 + 45(n - 3)}\) but make algebraic errors when expanding or combining like terms.

Common mistakes include getting \(\mathrm{180 + 45n - 45 = 45n + 135}\), which leads them to select Choice (C) (\(\mathrm{C(n) = 45n + 135}\)).

The Bottom Line:

This problem tests whether students can correctly interpret "additional beyond the first few" language and translate it into the mathematical expression \(\mathrm{(n - 3)}\). The key insight is understanding that if you have n users total and the first 3 are covered by the base price, then only \(\mathrm{(n - 3)}\) users incur the additional charge.