The value of a particular stock was $20 at the beginning of 2022. By the end of 2022, its value...

1. TRANSLATE the problem information

• Given information:
  • Initial stock value: \(\$20\)
  • Value increased by factor r at end of 2022
  • Value increased by same factor r again at end of 2023
  • Final value at end of 2023: \(\$45\)

• What "increased by a factor of r" means: multiply the current value by r

2. INFER the mathematical relationship

• Since the stock increases by factor r twice consecutively:
  • End of 2022: \(\$20 × \mathrm{r}\)
  • End of 2023: \((\$20 × \mathrm{r}) × \mathrm{r} = \$20\mathrm{r}^2\)

• This creates our key equation: \(20\mathrm{r}^2 = 45\)

3. SIMPLIFY to solve for r

• Divide both sides by 20:

\(\mathrm{r}^2 = 45/20 = 9/4 = 2.25\)

• Take the square root of both sides:

\(\mathrm{r} = \sqrt{2.25} = 1.5\) (use calculator if needed)

• Since we're dealing with a growth factor, we take the positive root

Answer: C (1.5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misinterpreting "increased by a factor of r" as addition instead of multiplication

Students might think: "The stock went from \(\$20\) to \(\$20 + \mathrm{r}\), then to \(\$20 + \mathrm{r} + \mathrm{r} = \$20 + 2\mathrm{r}\)"
This leads to the equation \(20 + 2\mathrm{r} = 45\), giving \(\mathrm{r} = 12.5\)

This approach doesn't match any answer choice, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Setting up the correct equation (\(20\mathrm{r}^2 = 45\)) but making calculation errors

Students might incorrectly calculate \(\mathrm{r}^2 = 45/20\) or make errors when taking the square root
For example, miscalculating \(\sqrt{2.25}\) could lead them toward Choice A (1.2) or Choice B (1.25)

The Bottom Line:

The key challenge is understanding that "factor" means multiplication, not addition. Growth factors create exponential relationships (\(\mathrm{r}^2\)) when applied consecutively, not linear ones (\(2\mathrm{r}\)).