Which of the following represents the result of increasing the quantity x by 9%, where x gt 0?

1. TRANSLATE the problem information

• Given information:
  • We need to increase quantity x by 9%
  • \(\mathrm{x \gt 0}\) (x is positive)
• What this tells us: We need to find the mathematical expression for x plus 9% of x

2. INFER the approach

• "Increase by 9%" means we add 9% of the original amount to the original amount
• This is different from just finding "9% of x" - we need the total after the increase

3. TRANSLATE the percentage to decimal form

• \(\mathrm{9\% = \frac{9}{100} = 0.09}\)
• So \(\mathrm{9\%\,of\,x = 0.09x}\)

4. INFER the final expression

• Original amount: \(\mathrm{x}\)
• Amount being added: \(\mathrm{0.09x}\)
• Total after increase: \(\mathrm{x + 0.09x = 1.09x}\)

Answer: A. \(\mathrm{1.09x}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse "increase by 9%" with "9% of x"

They see "9%" and immediately think the answer should be \(\mathrm{0.09x}\), forgetting that an increase means adding to the original amount. They're finding the amount of the increase rather than the final result after the increase.

This may lead them to select Choice B (\(\mathrm{0.09x}\))

Second Most Common Error:

Poor TRANSLATE reasoning: Students confuse percentage increase with absolute increase

They interpret "increase by 9%" as "increase by 9" and think the answer should be \(\mathrm{x + 9}\), mixing up percentage notation with regular numbers.

This may lead them to select Choice C (\(\mathrm{x + 9}\))

The Bottom Line:

The key insight is recognizing that "increase by a percentage" means you end up with more than \(\mathrm{100\%}\) of the original - specifically, you have \(\mathrm{100\%}\) + the percentage increase. So increasing by \(\mathrm{9\%}\) gives you \(\mathrm{109\% = 1.09}\) times the original.