From January 2012 to July 2014, the number of smartphone subscribers in a region increased from 1.5 million to 3.0...

1. TRANSLATE the problem information

• Given information:
  • January 2012: 1.5 million subscribers
  • July 2014: 3.0 million subscribers
  • Constant rate of increase
  • Need function \(\mathrm{S(m)}\) where \(\mathrm{m}\) = months after January 2012
• What this tells us: We have two data points and need to find a linear function

2. INFER the approach

• For a linear function \(\mathrm{S(m) = km + b}\), we need:
  • The y-intercept (b): value when \(\mathrm{m = 0}\)
  • The slope (k): rate of change
• January 2012 represents \(\mathrm{m = 0}\), so \(\mathrm{b = 1.5}\)
• We need to find how many months elapsed to July 2014

3. SIMPLIFY the time calculation

• From January 2012 to July 2014:
  • January 2012 to January 2014 = 2 years = 24 months
  • January 2014 to July 2014 = 6 months
  • Total: 24 + 6 = 30 months
• So our two points are: \(\mathrm{(0, 1.5)}\) and \(\mathrm{(30, 3.0)}\)

4. SIMPLIFY the slope calculation

• Using slope formula: \(\mathrm{k = \frac{y_2 - y_1}{x_2 - x_1}}\)
• \(\mathrm{k = \frac{3.0 - 1.5}{30 - 0}}\)
  \(\mathrm{= \frac{1.5}{30}}\)
  \(\mathrm{= \frac{1}{20}}\)

5. Build the final function

• \(\mathrm{S(m) = \frac{1}{20}m + 1.5}\)
• This matches choice (C)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret the time frame, calculating only 2.5 years instead of converting properly to months, or incorrectly determining what \(\mathrm{m = 0}\) represents.

Some students calculate the time as 2.5 years = 2.5 × 12 = 30 months correctly but then use the wrong reference point, thinking July 2014 corresponds to \(\mathrm{m = 0}\) instead of January 2012. This leads to incorrect slope calculations and may cause them to select Choice (D) (\(\mathrm{S(m) = \frac{1}{20}m + 3.0}\)) where they've switched the y-intercept.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the slope calculation but make arithmetic errors when simplifying 1.5/30, perhaps getting 1/30 instead of 1/20.

This leads them to select Choice (B) (\(\mathrm{S(m) = \frac{1}{30}m + 1.5}\)), which has the correct y-intercept but wrong slope.

The Bottom Line:

This problem tests whether students can systematically translate a real-world scenario into mathematical coordinates and then apply the linear function formula correctly. The key challenge is maintaining accuracy through multiple calculation steps while keeping track of what each variable represents.