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

1. TRANSLATE the problem information

• Given information:
  • Population increased by \(7\%\) from 2015 to 2016
  • \(\mathrm{2016\ population} = \mathrm{k} \times \mathrm{2015\ population}\)
• We need to find the value of k

2. INFER the mathematical relationship

• When something increases by \(7\%\), the new amount isn't just the original plus \(7\%\)
• The new amount is \(100\% + 7\% = 107\%\) of the original
• This means: \(\mathrm{2016\ population} = 1.07 \times \mathrm{2015\ population}\)

3. TRANSLATE this insight into an equation

• We have: \(\mathrm{2016\ population} = \mathrm{k} \times \mathrm{2015\ population}\)
• We also know: \(\mathrm{2016\ population} = 1.07 \times \mathrm{2015\ population}\)
• Therefore: \(\mathrm{k} \times \mathrm{2015\ population} = 1.07 \times \mathrm{2015\ population}\)

4. SIMPLIFY to find k

• Since both sides equal 2015 population times something, we can divide both sides by 2015 population
• This gives us: \(\mathrm{k} = 1.07\)

Answer: C. 1.07

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the increase amount with the multiplier

Many students see "increased by \(7\%\)" and immediately think \(\mathrm{k} = 0.07\), focusing only on the percentage increase rather than the total multiplier. They're thinking about how much was added (\(7\% = 0.07\)) instead of what the new total represents (\(107\% = 1.07\) times the original).

This leads them to select Choice A (0.07)

Second Most Common Error:

Conceptual confusion about percentages: Students misread \(7\%\) as \(70\%\)

Some students might misinterpret the percentage, thinking the population increased by \(70\%\) instead of \(7\%\). Following correct logic but with wrong input, they would get \(\mathrm{k} = 1.70\).

This may lead them to select Choice D (1.7)

The Bottom Line:

This problem tests whether students understand that a \(7\%\) increase means the new value is \(107\%\) (or \(1.07\) times) the original value, not just the \(7\%\) increase alone. The key insight is distinguishing between the amount of increase versus the total multiplier.