The combined price for a textbook and a scientific calculator before sales tax is $45. The textbook is subject to...

1. TRANSLATE the problem information

• Given information:
  • Combined pre-tax price: \(\$45\)
  • Textbook tax rate: \(6\%\)
  • Calculator tax rate: \(8\%\)
  • Combined after-tax price: \(\$48.00\)

• Let \(\mathrm{t}\) = pre-tax price of textbook, \(\mathrm{c}\) = pre-tax price of calculator

2. TRANSLATE the first equation

• The pre-tax total gives us: \(\mathrm{t + c = 45}\)

3. INFER how sales tax works

• Sales tax is added to the original price
• After-tax price = Original price + (Tax rate × Original price)
• This means: After-tax price = Original price × (1 + Tax rate)

4. TRANSLATE the after-tax prices

• Textbook after tax: \(\mathrm{t \times (1 + 0.06) = 1.06t}\)
• Calculator after tax: \(\mathrm{c \times (1 + 0.08) = 1.08c}\)

5. TRANSLATE the second equation

• Combined after-tax price: \(\mathrm{1.06t + 1.08c = 48.00}\)

The system is: \(\mathrm{t + c = 45}\) and \(\mathrm{1.06t + 1.08c = 48.00}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students use only the tax percentages (0.06 and 0.08) instead of the total multipliers (1.06 and 1.08) in their second equation.

They think: "6% tax means multiply by 0.06" rather than understanding that the after-tax price includes both the original price AND the tax. This leads to the equation \(\mathrm{0.06t + 0.08c = 48.00}\).

This may lead them to select Choice A (\(\mathrm{t + c = 45}\); \(\mathrm{0.06t + 0.08c = 48.00}\))

Second Most Common Error:

Poor TRANSLATE execution: Students correctly understand the tax multiplier concept but mix up which tax rate goes with which item, writing \(\mathrm{1.08t + 1.06c = 48.00}\) instead of \(\mathrm{1.06t + 1.08c = 48.00}\).

This may lead them to select Choice B (\(\mathrm{t + c = 45}\); \(\mathrm{1.08t + 1.06c = 48.00}\))

The Bottom Line:

Success requires careful attention to what sales tax actually means (you pay the original price PLUS the tax) and precise tracking of which tax rate applies to which item.