Hiro and Sofia purchased shirts and pants from a store. The price of each shirt purchased was the same and...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{x}\) = price of each shirt (in dollars)
  • \(\mathrm{y}\) = price of each pair of pants (in dollars)
  • Hiro: bought 4 shirts + 2 pairs of pants for $86 total
  • Sofia: bought 3 shirts + 5 pairs of pants for $166 total

2. INFER the approach

• Each person's purchases create one equation in our system
• We need to express: (number of shirts × price per shirt) + (number of pants × price per pants) = total cost

3. TRANSLATE Hiro's purchases into an equation

• Hiro spent: \(\mathrm{4x + 2y = 86}\)
  • 4 shirts at \(\mathrm{x}\) dollars each: \(\mathrm{4x}\)
  • 2 pairs of pants at \(\mathrm{y}\) dollars each: \(\mathrm{2y}\)
  • Total: $86

4. TRANSLATE Sofia's purchases into an equation

• Sofia spent: \(\mathrm{3x + 5y = 166}\)
  • 3 shirts at \(\mathrm{x}\) dollars each: \(\mathrm{3x}\)
  • 5 pairs of pants at \(\mathrm{y}\) dollars each: \(\mathrm{5y}\)
  • Total: $166

5. Form the system

• The system of equations is:
  \(\mathrm{4x + 2y = 86}\)
  \(\mathrm{3x + 5y = 166}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students mix up the quantities each person purchased while correctly identifying the costs.

For example, they might think Sofia bought 4 shirts and Hiro bought 3 shirts, leading to:
\(\mathrm{4x + 3y = 86}\)
\(\mathrm{2x + 5y = 166}\)

This may lead them to select Choice B.

Second Most Common Error:

Poor TRANSLATE reasoning: Students mix up which person spent which total amount while correctly identifying quantities.

They might write the equations as:
\(\mathrm{4x + 2y = 166}\) (thinking Hiro spent $166)
\(\mathrm{3x + 5y = 86}\) (thinking Sofia spent $86)

This may lead them to select Choice C.

The Bottom Line:

This problem requires careful reading and systematic translation of each person's shopping details. Students must track three pieces of information per person: quantities of each item and total cost. The mathematical concepts are straightforward, but accuracy in translating the scenario is crucial.