Sarah purchased a laptop by making an initial payment of $86 plus m monthly installments of $42 each. The total...

1. TRANSLATE the problem information

• Given information:
  • Initial payment: \(\$86\) (one-time payment)
  • Monthly installments: \(\mathrm{m}\) payments of \(\$42\) each
  • Total amount paid: \(\$632\)
• What this tells us: We have two parts making up the total cost

2. INFER the relationship

• The total amount Sarah paid consists of two components added together:
  • The initial payment (\(\$86\))
  • All the monthly payments combined (\(\$42 \times \mathrm{m} = \$42\mathrm{m}\))
• Therefore: Total = Initial payment + Monthly installments

3. TRANSLATE into equation form

• Set up the equation: \(\$86 + \$42\mathrm{m} = \$632\)
• This can be written as: \(\$42\mathrm{m} + \$86 = \$632\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students misinterpret the phrase "initial payment of $86 plus m monthly installments" and think they need to subtract somewhere, perhaps viewing the initial payment as reducing the total owed rather than contributing to it.

This leads them to set up equations like \(\$42\mathrm{m} - \$86 = \$632\), thinking the monthly payments minus the initial payment equals the total.

This may lead them to select Choice A (\(42\mathrm{m} - 86 = 632\)).

Second Most Common Error:

Poor TRANSLATE execution: Students correctly recognize they need addition but swap the coefficients, thinking there are 86 monthly payments of $42 each rather than m monthly payments of $42 each.

This confusion leads to the equation \(\$86\mathrm{m} + \$42 = \$632\).

This may lead them to select Choice D (\(86\mathrm{m} + 42 = 632\)).

The Bottom Line:

This problem requires careful reading to distinguish between the one-time initial payment and the repeated monthly payments, then correctly translating that structure into algebraic form.