QUESTION STEM:A mobile data plan charges a fixed monthly fee plus a constant rate per gigabyte (GB) of data used....

1. TRANSLATE the problem information

• Given information:
  • Fixed monthly fee (unknown) + constant rate per GB (unknown)
  • January: 8 GB used, total charge = $94
  • February: 12 GB used, total charge = $126

• What we need to find: The fixed monthly fee in dollars

2. INFER the mathematical approach

• This describes a linear relationship: \(\mathrm{Total\,Cost = Fixed\,Fee + (Rate × GB\,used)}\)
• We have two different scenarios, which gives us two equations with two unknowns
• We can set up a system of linear equations to solve for both the fixed fee and the rate

3. TRANSLATE into mathematical equations

• Let f = fixed monthly fee (dollars)
• Let r = rate per GB (dollars per GB)

• From January data: \(\mathrm{f + 8r = 94}\)
• From February data: \(\mathrm{f + 12r = 126}\)

4. INFER the solution strategy

• Since we have two equations with two unknowns, we can use elimination or substitution
• Elimination looks efficient here since both equations have the same f term

5. SIMPLIFY using elimination method

• Subtract the first equation from the second:
  \(\mathrm{(f + 12r) - (f + 8r) = 126 - 94}\)
• This gives us:
  \(\mathrm{f + 12r - f - 8r = 32}\)
• Simplifying:
  \(\mathrm{4r = 32}\)
• Therefore:
  \(\mathrm{r = 8}\) dollars per GB

6. SIMPLIFY to find the fixed fee

• Substitute \(\mathrm{r = 8}\) back into the first equation:
  \(\mathrm{f + 8(8) = 94}\)
• This gives us:
  \(\mathrm{f + 64 = 94}\)
• Therefore:
  \(\mathrm{f = 30}\) dollars

7. APPLY CONSTRAINTS and verify

• Check our solution against both original conditions:
  • January: \(\mathrm{30 + 8(8) = 30 + 64 = 94}\) ✓
  • February: \(\mathrm{30 + 12(8) = 30 + 96 = 126}\) ✓

Answer: 30

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may struggle to set up the correct equations from the word problem. They might write something like \(\mathrm{8 + r = 94}\) or \(\mathrm{f + 8 = 94}\), missing that the total cost includes both the fixed fee AND the usage charges.

This leads to confusion about what the variables represent and makes systematic solution impossible, causing them to guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equations but make algebraic errors during elimination or substitution. A common mistake is incorrectly distributing the negative sign when subtracting equations, getting \(\mathrm{-4r = 32}\) instead of \(\mathrm{4r = 32}\).

This leads them to calculate \(\mathrm{r = -8}\), which then gives \(\mathrm{f = 94 - 8(-8) = 158}\). They might mistakenly think the fixed fee is $158.

The Bottom Line:

This problem tests whether students can recognize a linear system hiding within a real-world rate structure. The key insight is that each monthly bill equals the same fixed fee plus a variable usage charge—once you see this pattern, the algebra becomes straightforward.