A company offers a subscription where the total cost \(\mathrm{C(n)}\), in dollars, for n months consists of a fixed sign-up...

1. TRANSLATE the problem information

• Given information:
  • Total cost \(\mathrm{C(n)}\) is linear in \(\mathrm{n}\) (number of months)
  • \(\mathrm{C(n)}\) includes a fixed sign-up fee plus constant monthly charge
  • \(\mathrm{C(3) = 41}\) and \(\mathrm{C(9) = 101}\)
  • Need to find the sign-up fee

• What this tells us: We need the linear function \(\mathrm{C(n) = mn + b}\), where \(\mathrm{m}\) is monthly charge and \(\mathrm{b}\) is the sign-up fee we're looking for.

2. INFER the solution strategy

• Since we have two points on a line, we can find both \(\mathrm{m}\) and \(\mathrm{b}\)
• Strategy: Use the two given points to create a system of equations, solve for slope first, then find the y-intercept

3. Set up the system of equations

From \(\mathrm{C(n) = mn + b}\):

• \(\mathrm{C(3) = 3m + b = 41}\)
• \(\mathrm{C(9) = 9m + b = 101}\)

4. SIMPLIFY to find the monthly charge

Subtract the first equation from the second to eliminate \(\mathrm{b}\):

\(\mathrm{(9m + b) - (3m + b) = 101 - 41}\)
\(\mathrm{6m = 60}\)
\(\mathrm{m = 10}\)

5. SIMPLIFY to find the sign-up fee

Substitute \(\mathrm{m = 10}\) into either equation. Using \(\mathrm{C(3) = 41}\):

\(\mathrm{3(10) + b = 41}\)
\(\mathrm{30 + b = 41}\)
\(\mathrm{b = 11}\)

Answer: D (11)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not properly set up the linear function or may confuse which variable represents what. They might write something like \(\mathrm{C(n) = bn + m}\), mixing up the roles of the slope and y-intercept.

This confusion about the structure leads them to solve for the wrong variable, potentially selecting Choice A (7) or Choice B (9) if they accidentally solve for a different quantity.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the system but make algebraic errors when solving. Common mistakes include arithmetic errors when subtracting equations or substitution errors when finding \(\mathrm{b}\).

For example, they might get \(\mathrm{m = 10}\) correctly but then make an error like \(\mathrm{41 - 30 = 10}\) instead of 11, leading them to select Choice C (10).

The Bottom Line:

This problem tests whether students can translate a real-world linear relationship into proper mathematical form and then systematically solve the resulting system. The key insight is recognizing that the 'sign-up fee' is the y-intercept of the linear function.