Ava loads $36 onto a laundry card. Each wash cycle deducts $1.75 from the card, and no other fees apply....

1. TRANSLATE the problem information

• Given information:
  • Initial balance: \(\$36\)
  • Cost per wash cycle: \(\$1.75\)
  • No other fees apply
  • Need to find balance B after w wash cycles

2. INFER the mathematical relationship

• Since each cycle removes money from the card, the balance decreases
• After w cycles, total amount removed = \(\mathrm{1.75 \times w = 1.75w}\)
• Remaining balance = Starting amount - Total removed

3. TRANSLATE into equation form

• \(\mathrm{B = 36 - 1.75w}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misunderstand the direction of the transaction, thinking that using the card somehow adds money rather than subtracting it.

They incorrectly reason: "Each cycle involves \(\$1.75\), so the balance changes by \(+\$1.75\) per cycle."

This may lead them to select Choice C (\(\mathrm{B = 36 + 1.75w}\))

Second Most Common Error:

Poor INFER skill: Students correctly identify that money is being subtracted but get confused about which quantities should be multiplied together.

They might think: "The initial amount should somehow be multiplied by the number of cycles" or misapply the distributive property.

This may lead them to select Choice B (\(\mathrm{B = 1.75(36 - w)}\)) or Choice D (\(\mathrm{B = 36w - 1.75}\))

The Bottom Line:

This problem tests whether students can correctly model a decreasing linear relationship. The key insight is recognizing that repeated equal deductions create a pattern where the total deduction equals (cost per cycle) × (number of cycles).