Question:A specialized printing machine produces business cards at a constant rate of 15 cards every 6 minutes. Working at this...

1. TRANSLATE the problem information

• Given information:
  • Machine produces 15 cards every 6 minutes (constant rate)
  • Need to find production in 1.5 hours
  • Answer must be complete cards (integer)

2. INFER the solution approach

• To find total production, we need: rate × time
• But first we need a consistent rate (cards per minute) and consistent time units
• Strategy: Convert to cards per minute, then convert 1.5 hours to minutes

3. SIMPLIFY to find the rate per minute

• Rate = \(15 \div 6 = 2.5\) cards per minute

4. TRANSLATE time to consistent units

• \(1.5\) hours = \(1.5 \times 60 = 90\) minutes

5. SIMPLIFY to find total production

• Total cards = \(2.5 \times 90 = 225\) cards

Answer: 225

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may try to work directly with mixed time units, attempting something like '15 cards per 6 minutes, so in 1.5 hours...' without establishing a clear rate per standard time unit.

This leads to confusion about how to handle the relationship between minutes and hours, often resulting in incorrect setups like trying to multiply \(15 \times 1.5\) directly, which gives \(22.5\) and doesn't make sense in context.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the need to find cards per minute (\(15 \div 6\)) but make arithmetic errors, getting 2 or 3 instead of 2.5, or they correctly find 2.5 but then make errors in the final multiplication (\(2.5 \times 90\)).

These arithmetic mistakes lead to answers like 180 or 270 instead of 225.

The Bottom Line:

This problem tests whether students can systematically work with rates by establishing consistent units before calculating totals. The key insight is recognizing that 'constant rate' problems require finding a unit rate first, then scaling up to the desired time period.