The equation y = 0.1x models the relationship between the number of different pieces of music a certain pianist practices,...

1. TRANSLATE the problem information

• Given information:
  • Equation: \(\mathrm{y = 0.1x}\)
  • \(\mathrm{y}\) = number of different pieces of music practiced
  • \(\mathrm{x}\) = number of minutes in practice session
  • The session lasted 30 minutes
• We need to find: How many pieces were practiced (the value of \(\mathrm{y}\))

2. INFER the approach

• Since we know \(\mathrm{x = 30}\) minutes and have the equation \(\mathrm{y = 0.1x}\), we can substitute the known value of \(\mathrm{x}\) to find \(\mathrm{y}\)
• Strategy: Substitute \(\mathrm{x = 30}\) into the equation and solve for \(\mathrm{y}\)

3. SIMPLIFY by performing the substitution and calculation

• Substitute \(\mathrm{x = 30}\) into \(\mathrm{y = 0.1x}\):

\(\mathrm{y = 0.1(30)}\)

• Calculate: \(\mathrm{y = 3}\)

Answer: B. 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor TRANSLATE reasoning: Students confuse which variable represents what quantity or don't recognize that they need to substitute the given time value into the equation.

Some students see "30 minutes" and think that must be the answer since it's the most prominent number, leading them to select Choice D (30) without performing any calculation.

Second Most Common Error:

Weak INFER skill: Students don't recognize the substitution strategy and instead try to manipulate the equation in unnecessary ways, or they misinterpret what 0.1 represents in the context.

This confusion about the decimal coefficient might lead them to think 0.1 means "1 out of 10" and select Choice C (10), or they might perform incorrect operations that yield Choice A (1).

The Bottom Line:

This problem tests whether students can connect the abstract mathematical relationship (\(\mathrm{y = 0.1x}\)) with concrete values from the problem context. The key insight is recognizing that "30 minutes" gives us the value of \(\mathrm{x}\), and we use the equation to find the corresponding value of \(\mathrm{y}\).