The number of zebras in a population in 2018 was 1.27 times the number of zebras in this population in...

1. TRANSLATE the problem information

• Given information:
  • 2018 zebra population = \(1.27 \times (2014\text{ zebra population})\)
  • 2014 zebra population = \(\mathrm{p\%}\) of 2018 zebra population
  • Need to find p to nearest whole number
• Let \(\mathrm{x}\) = zebras in 2014, \(\mathrm{y}\) = zebras in 2018

2. TRANSLATE each relationship into equations

• From "2018 was 1.27 times 2014": \(\mathrm{y = 1.27x}\)
• From "2014 is p% of 2018": \(\mathrm{x = (p/100)y}\)

3. INFER the solving strategy

• We have two equations with three unknowns (x, y, p)
• Key insight: We can substitute one equation into the other to eliminate x and y
• This will leave us with an equation we can solve for p

4. SIMPLIFY by substitution

• Substitute \(\mathrm{y = 1.27x}\) into \(\mathrm{x = (p/100)y}\):
  \(\mathrm{x = (p/100)(1.27x)}\)
• Divide both sides by x:
  \(\mathrm{1 = (p \times 1.27)/100}\)
• Multiply both sides by 100:
  \(\mathrm{100 = p \times 1.27}\)
• Divide by 1.27:
  \(\mathrm{p = 100/1.27 = 78.74...}\) (use calculator)

5. APPLY CONSTRAINTS for final answer

• Round to nearest whole number: \(\mathrm{p = 79}\)

Answer: 79

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often confuse the direction of the percentage relationship. They might write 2018 as p% of 2014 instead of 2014 as p% of 2018, leading to the equation \(\mathrm{y = (p/100)x}\) instead of \(\mathrm{x = (p/100)y}\).

With this incorrect setup, they would solve \(\mathrm{p \times 1.27 = 100}\), getting \(\mathrm{p = 100/1.27 = 127}\), which seems reasonable but is wrong. This leads to confusion and guessing among the higher answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equations but make algebraic errors during substitution. They might incorrectly manipulate \(\mathrm{x = (p/100)(1.27x)}\) and fail to properly cancel the x terms, leading to equations that don't solve cleanly.

This causes them to get stuck with messy expressions and resort to guessing.

The Bottom Line:

This problem tests whether students can correctly interpret bidirectional relationships (A is 1.27 times B means B is 1/1.27 times A) and translate percentage language precisely. The algebra is straightforward once the setup is correct.