A subscription service charges a one-time sign-up fee of f dollars and a constant fee of k dollars per month....

1. TRANSLATE the problem information

• Given equation: \(\frac{\mathrm{A - f}}{\mathrm{t}} = \mathrm{k}\)
• Need to find: \(\mathrm{A - f}\) expressed in terms of \(\mathrm{k}\) and \(\mathrm{t}\)
• What this tells us: We have \(\mathrm{A - f}\) divided by \(\mathrm{t}\) equals \(\mathrm{k}\), so we need to "undo" the division

2. SIMPLIFY to isolate the target expression

• Since \(\frac{\mathrm{A - f}}{\mathrm{t}} = \mathrm{k}\), we can multiply both sides by \(\mathrm{t}\)
• Left side: \(\frac{\mathrm{A - f}}{\mathrm{t}} \times \mathrm{t} = \mathrm{A - f}\)
• Right side: \(\mathrm{k} \times \mathrm{t} = \mathrm{kt}\)
• Result: \(\mathrm{A - f} = \mathrm{kt}\)

Answer: D (\(\mathrm{kt}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students get confused about which algebraic operation to apply

Instead of multiplying both sides by \(\mathrm{t}\), they might:

• Divide both sides by \(\mathrm{t}\), getting \(\mathrm{A - f} = \frac{\mathrm{k}}{\mathrm{t}}\)
• Think they need to multiply \(\mathrm{k}\) by \(\mathrm{t}\) and divide by something else

This may lead them to select Choice A (\(\frac{\mathrm{k}}{\mathrm{t}}\)) or get confused about the relationship between the variables.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret what the equation represents or what they're looking for

They might think:

• The equation means something different than it does
• They need to find \(\mathrm{A}\) instead of \(\mathrm{A - f}\)
• The variables should be combined differently (like \(\mathrm{k + t}\))

This may lead them to select Choice C (\(\mathrm{k + t}\)) or causes them to get stuck and guess.

The Bottom Line:

This problem tests whether students can recognize that to isolate \(\mathrm{A - f}\) from a fraction, they need to multiply both sides by the denominator. The key insight is seeing that \(\frac{\mathrm{A - f}}{\mathrm{t}} = \mathrm{k}\) is just asking "what divided by \(\mathrm{t}\) gives \(\mathrm{k}\)?" and the answer is \(\mathrm{kt}\).