A catering company charges a flat setup fee of $30 plus $15 per guest for a private event. Which function...

1. TRANSLATE the problem information

• Given information:
  • Setup fee: $30 (fixed cost, paid once regardless of guest count)
  • Cost per guest: $15 (variable cost, multiplied by number of guests)
  • Need to find: Function \(\mathrm{C(n)}\) for total cost with \(\mathrm{n}\) guests

2. INFER the mathematical structure

• This is a linear cost function with two components:
  • Fixed cost (doesn't change with guest count)
  • Variable cost (scales with guest count)
• Total cost = Fixed cost + (Variable cost × Number of units)

3. TRANSLATE into function notation

• Fixed cost becomes the constant term: +30
• Variable cost becomes the coefficient of \(\mathrm{n}\): \(\mathrm{15n}\)
• Therefore: \(\mathrm{C(n) = 15n + 30}\)

4. Verify by checking the structure

• When \(\mathrm{n = 0}\): \(\mathrm{C(0) = 15(0) + 30 = 30}\) (just the setup fee ✓)
• When \(\mathrm{n = 1}\): \(\mathrm{C(1) = 15(1) + 30 = 45}\) (setup fee + one guest ✓)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse which number goes with which cost component, swapping the coefficients.

They might think: "There are 30 guests paying $15 each, plus a $15 setup fee," misreading the problem structure. This leads them to write \(\mathrm{C(n) = 30n + 15}\), putting the larger number (30) as the per-guest cost.

This may lead them to select Choice B (\(\mathrm{30n + 15}\)).

Second Most Common Error:

Poor INFER reasoning about function structure: Students don't recognize that the setup fee should be a constant term, instead thinking both costs scale with the number of guests.

They might distribute incorrectly, writing \(\mathrm{C(n) = 15(n + 30)}\) or \(\mathrm{C(n) = 30(n + 15)}\), making both the setup fee and guest cost depend on \(\mathrm{n}\).

This may lead them to select Choice C (\(\mathrm{15(n + 30)}\)) or Choice D (\(\mathrm{30(n + 15)}\)).

The Bottom Line:

Success requires careful attention to which costs are fixed versus variable. The setup fee happens once (constant term), while the per-guest cost repeats \(\mathrm{n}\) times (coefficient of \(\mathrm{n}\)). Students who rush through the TRANSLATE step often swap these components.