The total mass, in kilograms, of r identical objects is t. Which expression represents the total mass, in kilograms, of...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{r}\) identical objects have total mass = \(\mathrm{t}\) kilograms
  • Want: total mass of \(\mathrm{146r}\) identical objects

• This tells us we're comparing two scenarios with different numbers of the same objects

2. INFER the relationship between quantities

• Key insight: \(\mathrm{146r}\) means \(\mathrm{146 \times r}\) objects
• We have \(\mathrm{146}\) times as many objects as the original scenario
• Since objects are identical, more objects means proportionally more mass

3. Apply the proportional relationship

• Original: \(\mathrm{r}\) objects → \(\mathrm{t}\) kilograms total mass
• New scenario: \(\mathrm{146r}\) objects → ? kilograms total mass
• Since \(\mathrm{146r = 146 \times r}\), we have \(\mathrm{146}\) times as many objects
• Therefore: \(\mathrm{Total\ mass = 146 \times t = 146t}\) kilograms

Answer: D. \(\mathrm{146t}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "\(\mathrm{146r}\) of these objects" as meaning "\(\mathrm{146}\) more objects than \(\mathrm{r}\)" rather than "\(\mathrm{146}\) times \(\mathrm{r}\) objects"

This leads them to think about adding quantities rather than scaling proportionally. They might reason: "We have \(\mathrm{r}\) objects with mass \(\mathrm{t}\), and we're adding \(\mathrm{146}\) somehow, so maybe the answer involves adding \(\mathrm{146}\) to something." This may lead them to select Choice B (\(\mathrm{146 + t}\)).

Second Most Common Error:

Poor INFER reasoning about proportional relationships: Students recognize that \(\mathrm{146r}\) means \(\mathrm{146}\) times as many objects, but incorrectly think this means dividing the mass rather than multiplying it.

They might think: "We have \(\mathrm{146}\) times as many objects, so each object has less mass, so we divide." This may lead them to select Choice C (\(\mathrm{t/146}\)).

The Bottom Line:

This problem tests whether students can recognize proportional scaling relationships and correctly translate between verbal descriptions and mathematical operations. The key insight is that "\(\mathrm{146}\) times as many identical objects" means "\(\mathrm{146}\) times the total mass."