The population of City A increased by 7% from 2015 to 2016. If the 2016 population is k times the...

1. TRANSLATE the problem information

• Given information:
  • Population increased by \(7\%\) from 2015 to 2016
  • 2016 population = \(\mathrm{k}\) × 2015 population
  • Need to find \(\mathrm{k}\)
• What this tells us: We need to express the relationship between the two populations mathematically.

2. TRANSLATE what "increased by 7%" means

• When something increases by \(7\%\), the new value equals:
  • Original value + \(7\%\) of original value
  • \(\mathrm{Original\ value} + 0.07 \times \mathrm{original\ value}\)
  • \((1 + 0.07) \times \mathrm{original\ value} = 1.07 \times \mathrm{original\ value}\)

3. INFER the relationship between k and the percentage increase

• We know: \(\mathrm{2016\ population} = \mathrm{k} \times \mathrm{2015\ population}\)
• From our translation: \(\mathrm{2016\ population} = 1.07 \times \mathrm{2015\ population}\)
• Therefore: \(\mathrm{k} \times \mathrm{2015\ population} = 1.07 \times \mathrm{2015\ population}\)
• This means: \(\mathrm{k} = 1.07\)

Answer: C. 1.07

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse "increased by 7%" with "is 7% of the original"

When students see "increased by 7%", they might think this means the 2016 population is only 7% of the 2015 population, leading them to conclude \(\mathrm{k} = 0.07\). This fundamental misunderstanding of percentage increase language causes them to miss that an increase means adding to 100%, not replacing it.

This leads them to select Choice A (0.07).

Second Most Common Error:

Incomplete TRANSLATE reasoning: Students partially understand the increase but make conversion errors

Some students correctly recognize that a 7% increase means more than the original, but they might think "7% more" simply means \(\mathrm{k} = 7\) or \(\mathrm{k} = 0.7\), forgetting to add this increase to the original 100%. They get confused about whether to use 7, 0.7, or some other form.

This may lead them to select Choice B (0.7) or cause confusion leading to guessing.

The Bottom Line:

The key challenge is correctly TRANSLATING percentage increase language into mathematical relationships. Students must understand that "increased by x%" means "is now \((100 + \mathrm{x})\%\) of the original value."