A movie theater charges $11 for each full-price ticket and $8.25 for each reduced-price ticket. For one movie showing, the...
GMAT Algebra : (Alg) Questions
A movie theater charges \(\$11\) for each full-price ticket and \(\$8.25\) for each reduced-price ticket. For one movie showing, the theater sold a total of \(214\) full-price and reduced-price tickets for \(\$2{,}145\). Which of the following systems of equations could be used to determine the number of full-price tickets, \(\mathrm{f}\), and the number of reduced-price tickets, \(\mathrm{r}\), sold?
\(\mathrm{11f + 8.25r = 214}\)
\(\mathrm{11f + 8.25r = 2{,}145}\)
\(\mathrm{8.25f + 11r = 2{,}145}\)
\(\mathrm{8.25f + 11r = 214}\)
1. TRANSLATE the problem information
- Given information:
- Full-price tickets cost $11 each (f tickets sold)
- Reduced-price tickets cost $8.25 each (r tickets sold)
- Total tickets sold: 214
- Total revenue: $2,145
2. INFER what relationships we need to capture
- We need two equations because we have two unknowns (f and r)
- First relationship: total quantity (tickets)
- Second relationship: total value (revenue)
3. TRANSLATE the first relationship into an equation
- Total tickets = full-price tickets + reduced-price tickets
- \(\mathrm{f + r = 214}\)
4. TRANSLATE the second relationship into an equation
- Total revenue = (price per full-price ticket × number of full-price tickets) + (price per reduced-price ticket × number of reduced-price tickets)
- \(\mathrm{11f + 8.25r = 2{,}145}\)
5. INFER which answer choice matches our system
- Our system: \(\mathrm{f + r = 214}\) and \(\mathrm{11f + 8.25r = 2{,}145}\)
- This matches Choice B exactly
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students mix up which numbers represent totals versus unit prices, leading them to swap values incorrectly.
For example, they might think the $2,145 represents the number of tickets sold and 214 represents the total revenue. This confusion stems from not carefully tracking what each number in the problem represents before writing equations.
This may lead them to select Choice A (\(\mathrm{f + r = 2{,}145}\) and \(\mathrm{11f + 8.25r = 214}\)).
Second Most Common Error:
Poor TRANSLATE reasoning: Students correctly identify the total tickets and revenue values but mix up which price goes with which variable type.
They might reason that since "reduced-price" sounds like the main category, it should get the higher coefficient, leading to equations like \(\mathrm{f + r = 214}\) and \(\mathrm{8.25f + 11r = 2{,}145}\).
This may lead them to select Choice C (\(\mathrm{f + r = 214}\) and \(\mathrm{8.25f + 11r = 2{,}145}\)).
The Bottom Line:
Success requires careful attention to which specific numbers correspond to which concepts (quantities vs. prices, totals vs. unit values) before translating into mathematical notation.
\(\mathrm{11f + 8.25r = 214}\)
\(\mathrm{11f + 8.25r = 2{,}145}\)
\(\mathrm{8.25f + 11r = 2{,}145}\)
\(\mathrm{8.25f + 11r = 214}\)