\(\mathrm{M = 1{,}800(1.02)^t}\) The equation above models the number of members, M, of a gym t years after the gym...

1. TRANSLATE the problem information

• Given information:
  • Original model: \(\mathrm{M = 1,800(1.02)^t}\) where \(\mathrm{t}\) = years
  • Need: Model where time is measured in quarter years (q)

• What this tells us: We need to express the same exponential relationship using a different time unit.

2. INFER the time relationship

• Key insight: We must connect years (t) to quarter years (q)
• Since \(\mathrm{1\text{ year} = 4\text{ quarters}}\), then \(\mathrm{q\text{ quarters} = q/4\text{ years}}\)
• Therefore: \(\mathrm{t = q/4}\)

3. SIMPLIFY through substitution

• Substitute \(\mathrm{t = q/4}\) into the original equation:
  \(\mathrm{M = 1,800(1.02)^t}\) becomes \(\mathrm{M = 1,800(1.02)^{(q/4)}}\)

• This matches answer choice A exactly.

Answer: A. \(\mathrm{M = 1,800(1.02)^{(q/4)}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students struggle to establish the correct relationship between years and quarter years, often thinking that q quarters means 4q years instead of q/4 years.

They might reason: "If there are 4 quarters in a year, then q quarters must be 4q years," leading to \(\mathrm{t = 4q}\). This would give \(\mathrm{M = 1,800(1.02)^{(4q)}}\), which matches Choice B.

Second Most Common Error:

Conceptual confusion about time conversion: Students correctly identify that they need to convert quarter years but attempt to modify the growth rate instead of the time variable.

They think: "Since we're measuring in quarters, the growth rate per quarter must be different," and try to find a quarterly growth rate. This leads them toward Choice C or D where the base has been modified.

The Bottom Line:

This problem tests whether students understand that changing the time unit in an exponential model requires substituting the time relationship into the exponent, not modifying other parts of the equation.