For groups of 25 or more people, a museum charges $21 per person for the first 25 people and $14...
GMAT Algebra : (Alg) Questions
For groups of 25 or more people, a museum charges \(\$21\) per person for the first \(25\) people and \(\$14\) for each additional person. Which function \(\mathrm{f}\) gives the total charge, in dollars, for a tour group with \(\mathrm{n}\) people, where \(\mathrm{n} \geq 25\)?
1. TRANSLATE the problem information
- Given information:
- Groups of n people, where \(\mathrm{n \geq 25}\)
- First 25 people cost \(\$21\) per person
- Each additional person costs \(\$14\) per person
- Need to find function \(\mathrm{f(n)}\) for total charge
2. INFER the approach
- The key insight is that we have two different pricing rates
- Split the group into two parts: the first 25 people (at \(\$21\) each) and the remaining people (at \(\$14\) each)
- Calculate the cost for each part separately, then add them together
3. Calculate the cost for each part
- Cost for first 25 people: \(\mathrm{25 \times \$21 = \$525}\)
- Number of additional people: \(\mathrm{n - 25}\)
- Cost for additional people: \(\mathrm{(n - 25) \times \$14 = \$14(n - 25)}\)
4. Combine the costs
- Total cost: \(\mathrm{f(n) = 525 + 14(n - 25)}\)
5. SIMPLIFY the expression algebraically
- \(\mathrm{f(n) = 525 + 14(n - 25)}\)
- \(\mathrm{f(n) = 525 + 14n - 350}\)
- \(\mathrm{f(n) = 14n + 175}\)
Answer: A. \(\mathrm{f(n) = 14n + 175}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret the pricing structure and think all n people are charged at one rate instead of recognizing the tiered pricing system.
They might think all people are charged \(\$21\) each: \(\mathrm{f(n) = 21n}\), or all people are charged \(\$14\) each: \(\mathrm{f(n) = 14n}\). Neither of these expressions matches any answer choice exactly, leading to confusion and guessing.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly set up \(\mathrm{f(n) = 525 + 14(n - 25)}\) but make algebraic errors during expansion.
They might incorrectly expand to get \(\mathrm{f(n) = 525 + 14n + 350 = 14n + 875}\), or make sign errors. This doesn't match any answer choice and may lead them to select Choice B (\(\mathrm{f(n) = 14n + 525}\)) if they remember the 525 but forget the subtraction.
The Bottom Line:
This problem requires recognizing that different portions of the group are charged at different rates, then correctly translating this piecewise structure into algebra. The key breakthrough is seeing the group as "first 25" plus "additional people beyond 25."