John paid a total of $165 for a microscope by making a down payment of $37 plus p monthly payments...

1. TRANSLATE the payment information

• Given information:
  • Total amount paid: \(\$165\)
  • Down payment: \(\$37\) (paid once)
  • Monthly payments: \(\$16\) each month for \(\mathrm{p}\) months
• What this tells us: We need to account for both the one-time payment and the recurring payments.

2. INFER the relationship between payments

• The total amount paid consists of two parts:
  • The down payment (happens once)
  • All the monthly payments combined
• Since there are \(\mathrm{p}\) monthly payments of \(\$16\) each, the total from monthly payments is \(16\mathrm{p}\)

3. Set up the equation

• Total amount = Down payment + All monthly payments

\(\$165 = \$37 + 16\mathrm{p}\)

\(16\mathrm{p} + 37 = 165\)

Answer: C. 16p + 37 = 165

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse which number should be multiplied by \(\mathrm{p}\). They might think the down payment of \(\$37\) happens \(\mathrm{p}\) times instead of the monthly payment of \(\$16\).

This leads them to write: \(37\mathrm{p} + 16 = 165\), causing them to select Choice D (37p + 16 = 165).

Second Most Common Error:

Poor TRANSLATE reasoning: Students might subtract the down payment instead of adding it, thinking it reduces the total amount owed rather than being part of what was already paid.

This may lead them to select Choice A (16p - 37 = 165).

The Bottom Line:

This problem tests whether students can correctly identify which payment happens multiple times (the monthly payment) versus which happens once (the down payment), then properly combine them in an equation.