Provider A charges a $18 monthly fee plus $2.50 per gigabyte for video data and $1.50 per gigabyte for other...

1. TRANSLATE the problem information

• Given information:
  • Provider A: \(\$18 + \$2.50/\mathrm{GB\ video} + \$1.50/\mathrm{GB\ other} = \$60\)
  • Provider B: \(\$12 + \$4.50/\mathrm{GB\ video} + \$3.50/\mathrm{GB\ other} = \$94\)
  • Need to find: gigabytes of other data used
• Let \(\mathrm{v}\) = video data (GB), \(\mathrm{o}\) = other data (GB)
• This gives us two equations in two unknowns

2. INFER that we need to set up a system of linear equations

• Provider A equation: \(18 + 2.50\mathrm{v} + 1.50\mathrm{o} = 60\)
• Provider B equation: \(12 + 4.50\mathrm{v} + 3.50\mathrm{o} = 94\)
• Since we have two equations and two unknowns, we can solve using elimination

3. SIMPLIFY the equations by removing constants

• Provider A: \(2.50\mathrm{v} + 1.50\mathrm{o} = 42\) (subtract 18 from both sides)
• Provider B: \(4.50\mathrm{v} + 3.50\mathrm{o} = 82\) (subtract 12 from both sides)

4. INFER the best elimination strategy

• To avoid decimals, multiply the first equation by 2:
  • \(5\mathrm{v} + 3\mathrm{o} = 84\)
  • \(4.5\mathrm{v} + 3.5\mathrm{o} = 82\)

5. SIMPLIFY using elimination

• Multiply first equation by 4.5: \(22.5\mathrm{v} + 13.5\mathrm{o} = 378\)
• Multiply second equation by 5: \(22.5\mathrm{v} + 17.5\mathrm{o} = 410\)
• Subtract first from second: \(4\mathrm{o} = 32\)
• Therefore: \(\mathrm{o} = 8\)

6. APPLY CONSTRAINTS and verify

• Find v: \(5\mathrm{v} + 3(8) = 84\), so \(\mathrm{v} = 12\)
• Check Provider A: \(18 + 2.50(12) + 1.50(8) = 60\) ✓
• Check Provider B: \(12 + 4.50(12) + 3.50(8) = 94\) ✓

Answer: C (8)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often struggle to correctly set up the system of equations from the word problem. They might mix up which provider has which rates, or incorrectly handle the base monthly fees.

This confusion with the setup leads to incorrect equations, making the rest of the solution meaningless. Students might end up with answers that don't make sense when verified, causing them to abandon systematic solution and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Even with correct equations, students often make calculation errors during the elimination process, especially when working with decimal coefficients. They might incorrectly multiply equations or make arithmetic mistakes when combining terms.

This may lead them to select Choice A (4) or Choice D (10) based on calculation errors.

The Bottom Line:

This problem requires careful attention to detail in both setting up the equations and executing the algebraic steps. The key insight is recognizing that the same usage pattern results in different bills with different providers, creating a solvable system.