A photography studio charges $180 for a portrait session, which includes the first hour of shooting. For portrait sessions lasting...
GMAT Algebra : (Alg) Questions
A photography studio charges \(\$180\) for a portrait session, which includes the first hour of shooting. For portrait sessions lasting longer than one hour, there is an additional charge of \(\$45\) for each hour beyond the first hour. Which function \(\mathrm{f}\) represents the total cost, in dollars, for a portrait session lasting \(\mathrm{h}\) hours, where \(\mathrm{h} \geq 1\)?
1. TRANSLATE the cost structure into mathematical components
- Given information:
- Base cost: $180 (includes first hour of shooting)
- Additional hourly rate: $45 (for each hour beyond the first)
- Total session length: \(\mathrm{h}\) hours (where \(\mathrm{h ≥ 1}\))
2. INFER how to count additional hours
- Key insight: If a session lasts \(\mathrm{h}\) hours total, then the additional hours beyond the first hour = \(\mathrm{(h-1)}\) hours
- For example: 3-hour session = 1 base hour + 2 additional hours
3. Build the cost function
- Total cost = Base cost + Cost of additional hours
- \(\mathrm{f(h) = 180 + 45(h-1)}\)
4. SIMPLIFY the expression algebraically
- \(\mathrm{f(h) = 180 + 45(h-1)}\)
- \(\mathrm{f(h) = 180 + 45h - 45}\) [distribute the 45]
- \(\mathrm{f(h) = 45h + 135}\) [combine like terms: \(\mathrm{180 - 45 = 135}\)]
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students incorrectly assume that \(\mathrm{h}\) hours means \(\mathrm{h}\) additional hours beyond the base, rather than recognizing that \(\mathrm{(h-1)}\) represents additional hours.
They set up: \(\mathrm{f(h) = 180 + 45h}\)
This leads them to select Choice (D) (45h + 180) or get confused when this exact form isn't listed.
Second Most Common Error:
Poor TRANSLATE reasoning: Students misunderstand the base cost structure, thinking the $180 is separate from the first hour rather than inclusive of it.
They might think: first hour costs some amount, plus $180 base fee, plus $45 for additional hours. This creates confusion about how to structure the function and may lead them to select Choice (A) (45h) thinking they need a simpler relationship.
The Bottom Line:
This problem tests whether students can correctly interpret "includes the first hour" language and translate it into the mathematical insight that additional hours = total hours minus one.