3 more than 8 times a number x is equal to 83. Which equation represents this situation?

1. TRANSLATE the problem phrase by phrase

• Given phrase: "3 more than 8 times a number x is equal to 83"
• Break it down:
  • "8 times a number x" → \(\mathrm{8x}\)
  • "3 more than [8 times a number x]" → \(\mathrm{8x + 3}\)
  • "is equal to 83" → \(\mathrm{= 83}\)

2. INFER the correct structure

• The phrase "3 more than [something]" means: [something] + 3
• So "3 more than 8x" becomes: \(\mathrm{8x + 3}\)
• This gives us the complete equation: \(\mathrm{8x + 3 = 83}\)

3. TRANSLATE to verify against answer choices

• Our equation \(\mathrm{8x + 3 = 83}\) matches Choice D exactly

Answer: D. \(\mathrm{8x + 3 = 83}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "3 more than 8 times x" as "3 times 8 times x"

They think the "3 more" means to multiply by 3, leading to the expression \(\mathrm{(3)(8)x = 24x}\). This creates the equation \(\mathrm{24x = 83}\).

This may lead them to select Choice A (\(\mathrm{(3)(8)x = 83}\))

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly identify \(\mathrm{8x}\) but misplace where the "3 more" goes in the equation structure

They might think "\(\mathrm{8x}\) is equal to 83 plus 3" instead of "\(\mathrm{8x}\) plus 3 is equal to 83", creating the equation \(\mathrm{8x = 83 + 3}\).

This may lead them to select Choice B (\(\mathrm{8x = 83 + 3}\))

The Bottom Line:

The key challenge is carefully translating English word order into correct mathematical expressions. The phrase "3 more than [quantity]" always means [quantity] + 3, never 3 + [quantity] or 3 × [quantity].