Which expression is equivalent to \(16(\mathrm{x} + 15)\)?

1. INFER the mathematical operation needed

• Given: \(16(x + 15)\)
• This expression shows multiplication of 16 times the quantity \((x + 15)\)
• When we have a number multiplied by a sum in parentheses, we need to use the distributive property

2. SIMPLIFY by applying the distributive property

• Apply \(a(b + c) = ab + ac\):

\(16(x + 15) = 16 \cdot x + 16 \cdot 15\)

3. SIMPLIFY the arithmetic

• Calculate each term:
  • \(16 \cdot x = 16x\)
  • \(16 \cdot 15 = 240\)
• Combine: \(16x + 240\)

Answer: B. \(16x + 240\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make an arithmetic error and add instead of multiply when dealing with the constant term.

Instead of calculating \(16 \times 15 = 240\), they might think \(16 + 15 = 31\). This happens when students rush through the problem or confuse the operations they need to perform.

This may lead them to select Choice A (\(16x + 31\)).

Second Most Common Error:

Poor INFER reasoning: Students don't recognize that they need to apply the distributive property at all.

They see \(16(x + 15)\) and incorrectly think the answer is just \(16x + 15\), completely ignoring that the 16 needs to be distributed to both terms inside the parentheses.

This may lead them to select Choice D (\(16x + 15\)).

The Bottom Line:

This problem tests whether students can systematically apply the distributive property and perform accurate arithmetic. The key insight is recognizing that every term inside the parentheses must be multiplied by the factor outside.