A manufacturing company produces two types of electronic components: processors and memory chips. Each processor requires $8.50 in raw materials...

1. TRANSLATE the problem information

• Given information:
  • Each processor requires \(\$8.50\) in raw materials
  • Each memory chip requires \(\$6.20\) in raw materials
  • Total budget allocated: \(\$178.40\)
  • Variables: \(\mathrm{x}\) = processors, \(\mathrm{y}\) = memory chips

2. INFER the approach

• We need an equation that represents the total cost constraint
• The total cost of raw materials must equal the allocated budget
• This means: (cost per processor × number of processors) + (cost per memory chip × number of memory chips) = total budget

3. TRANSLATE each cost component

• Cost for x processors: \(\$8.50 \times \mathrm{x} = 8.5\mathrm{x}\) dollars
• Cost for y memory chips: \(\$6.20 \times \mathrm{y} = 6.2\mathrm{y}\) dollars

4. TRANSLATE the complete equation

• Total cost equation: \(8.5\mathrm{x} + 6.2\mathrm{y} = 178.4\)
• This matches choice (B)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students mix up which cost goes with which variable, reversing the coefficients to get \(6.2\mathrm{x} + 8.5\mathrm{y} = 178.4\).

They read "processors require $8.50" and "memory chips require $6.20" but then incorrectly assign the $8.50 to y (memory chips) and $6.20 to x (processors). This reversal happens when students don't carefully track which variable represents which product.

This may lead them to select Choice A \((6.2\mathrm{x} + 8.5\mathrm{y} = 178.4)\).

Second Most Common Error:

Poor INFER reasoning: Students fail to recognize that this is a cost constraint problem and instead think the equation should simply count the total number of items produced.

They ignore the individual costs entirely and assume the constraint is just on the total quantity: \(\mathrm{x} + \mathrm{y} = 178.4\). This happens when students don't understand that the $178.40 represents a budget limit on raw material costs, not a quantity limit.

This may lead them to select Choice C \((\mathrm{x} + \mathrm{y} = 178.4)\).

The Bottom Line:

This problem tests whether students can systematically convert real-world cost constraints into mathematical equations while carefully tracking which costs belong to which variables. Success requires methodical translation and clear understanding of what the constraint actually represents.