The total cost, in dollars, to produce n gadgets is given by the expression 2n + 30. If the total...

1. TRANSLATE the problem information

• Given information:
  • Cost expression: \(\mathrm{2n + 30}\) (where \(\mathrm{n}\) = number of gadgets)
  • Total cost: 70 dollars
  • Need to find: \(\mathrm{n + 15}\)
• This gives us the equation: \(\mathrm{2n + 30 = 70}\)

2. INFER the approach

• We need to solve for n first, then calculate \(\mathrm{n + 15}\)
• We can solve this using standard algebraic steps, or notice a clever shortcut

3. SIMPLIFY using standard algebra

• Start with: \(\mathrm{2n + 30 = 70}\)
• Subtract 30 from both sides: \(\mathrm{2n = 40}\)
• Divide both sides by 2: \(\mathrm{n = 20}\)
• Calculate what we need: \(\mathrm{n + 15 = 20 + 15 = 35}\)

Alternative clever approach:

• Notice that \(\mathrm{2n + 30 = 2(n + 15)}\)
• So our equation becomes: \(\mathrm{2(n + 15) = 70}\)
• Divide by 2: \(\mathrm{n + 15 = 35}\) (directly!)

Answer: B. 35

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students correctly solve \(\mathrm{2n + 30 = 70}\) to get \(\mathrm{n = 20}\), but then stop there and select 20 as their final answer, forgetting that the question asks for \(\mathrm{n + 15}\).

This leads them to select Choice A (20).

Second Most Common Error:

Poor final calculation: Students find \(\mathrm{n = 20}\) correctly, then get confused about what to do next. Instead of calculating \(\mathrm{n + 15 = 35}\), they might subtract n from the total cost (\(\mathrm{70 - 20 = 50}\)) or add both n and \(\mathrm{n + 15}\) together (\(\mathrm{20 + 35 = 55}\)).

This may lead them to select Choice C (50) or Choice D (55).

The Bottom Line:

This problem tests whether students can stay focused on what the question is actually asking for. The algebra is straightforward, but students must resist the temptation to stop once they find n, and instead complete the calculation to find \(\mathrm{n + 15}\).