2.5b + 5r = 80The given equation describes the relationship between the number of birds, b, and the number of...

1. TRANSLATE the problem information

• Given equation: \(2.5\mathrm{b} + 5\mathrm{r} = 80\)
• Given: The business cares for 16 reptiles, so \(\mathrm{r} = 16\)
• Find: The number of birds (b) the business can care for

2. INFER the solution approach

• Since we know \(\mathrm{r} = 16\), we can substitute this value into the equation and solve for b
• This substitution will give us a simple equation with only one unknown

3. SIMPLIFY by substituting and solving

• Substitute \(\mathrm{r} = 16\): \(2.5\mathrm{b} + 5(16) = 80\)
• Calculate: \(2.5\mathrm{b} + 80 = 80\)
• Subtract 80 from both sides: \(2.5\mathrm{b} = 0\)
• Divide by 2.5: \(\mathrm{b} = 0\)

Answer: A. 0

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors during the algebraic steps, particularly when calculating \(5(16) = 80\) or when subtracting 80 from both sides.

For example, they might incorrectly calculate \(5(16) = 50\), leading to \(2.5\mathrm{b} + 50 = 80\), which gives \(2.5\mathrm{b} = 30\), and \(\mathrm{b} = 12\). This could lead them to guess among the available choices or become confused since 12 isn't an option.

Second Most Common Error:

Poor TRANSLATE reasoning: Students might confuse which variable represents which quantity, substituting \(\mathrm{b} = 16\) instead of \(\mathrm{r} = 16\).

This leads to: \(2.5(16) + 5\mathrm{r} = 80\), giving \(40 + 5\mathrm{r} = 80\), so \(5\mathrm{r} = 40\), and \(\mathrm{r} = 8\). This doesn't directly answer the question about birds, causing confusion and potential guessing.

The Bottom Line:

This problem tests whether students can correctly identify what information they're given, substitute appropriately, and execute basic algebra without computational errors. The counterintuitive result (0 birds) makes it essential to trust the algebraic process rather than second-guessing the mathematical outcome.