The cost per student C for a field trip is represented by the equation C = (T + F)/n, where...

1. TRANSLATE the problem requirement

• Given equation: \(\mathrm{C = (T + F)/n}\)
• Need to find: \(\mathrm{T}\) expressed in terms of \(\mathrm{C, F, and\ n}\)
• What this means: Rearrange the equation so \(\mathrm{T}\) is alone on one side

2. SIMPLIFY by eliminating the fraction

• The equation has \(\mathrm{T + F}\) in the numerator, divided by \(\mathrm{n}\)
• To eliminate this fraction, multiply both sides by \(\mathrm{n}\):
  • Left side: \(\mathrm{C \times n = Cn}\)
  • Right side: \(\mathrm{[(T + F)/n] \times n = T + F}\)
• Result: \(\mathrm{Cn = T + F}\)

3. SIMPLIFY by isolating T

• Now we have: \(\mathrm{Cn = T + F}\)
• To get \(\mathrm{T}\) alone, subtract \(\mathrm{F}\) from both sides:
  • Left side: \(\mathrm{Cn - F}\)
  • Right side: \(\mathrm{T + F - F = T}\)
• Result: \(\mathrm{T = Cn - F}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors or forget proper algebraic steps when manipulating the equation.

For example, they might try to "move" \(\mathrm{F}\) to the other side without properly applying it to the entire equation, leading to incorrect expressions like \(\mathrm{T = C - F/n}\). Or they might add \(\mathrm{F}\) instead of subtract it in the final step, getting \(\mathrm{T = Cn + F}\).

This may lead them to select Choice B (\(\mathrm{T = C - F/n}\)) or Choice D (\(\mathrm{T = Cn + F}\))

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what "express \(\mathrm{T}\) in terms of \(\mathrm{C, F, and\ n}\)" means and attempt shortcuts without systematic algebraic manipulation.

They might try to rearrange terms without following proper algebraic rules, leading to expressions like \(\mathrm{(C - F)/n}\) where they incorrectly group terms.

This may lead them to select Choice C (\(\mathrm{T = (C - F)/n}\))

The Bottom Line:

This problem tests systematic algebraic thinking. Success requires methodically applying inverse operations in the correct order - first eliminating fractions, then isolating the target variable.