A student council group is selling school posters for a fundraiser. They use the function \(\mathrm{p(x) = 5x - 220}\)...
GMAT Algebra : (Alg) Questions
A student council group is selling school posters for a fundraiser. They use the function \(\mathrm{p(x) = 5x - 220}\) to determine their profit \(\mathrm{p(x)}\), in dollars, for selling \(\mathrm{x}\) school posters. In order to earn a profit of \($900\), how many school posters must they sell?
1. TRANSLATE the problem information
- Given information:
- Profit function: \(\mathrm{p(x) = 5x - 220}\)
- \(\mathrm{p(x)}\) = profit in dollars
- \(\mathrm{x}\) = number of posters sold
- Want to find \(\mathrm{x}\) when profit = $900
- What this tells us: We need to find the input value (\(\mathrm{x}\)) that produces an output value (\(\mathrm{p(x)}\)) of $900.
2. INFER the solution strategy
- Since we want a profit of $900, we need to set \(\mathrm{p(x) = 900}\)
- This gives us the equation: \(\mathrm{900 = 5x - 220}\)
- Now we solve this linear equation for \(\mathrm{x}\)
3. SIMPLIFY through algebraic manipulation
- Start with: \(\mathrm{900 = 5x - 220}\)
- Add 220 to both sides: \(\mathrm{900 + 220 = 5x}\)
- Calculate: \(\mathrm{1120 = 5x}\)
- Divide both sides by 5: \(\mathrm{x = 1120 ÷ 5 = 224}\)
Answer: 224 posters
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students may misunderstand what the question is asking. They might think they need to substitute 224 for \(\mathrm{x}\) in the original equation, or they might confuse which variable they're solving for. This fundamental misreading prevents them from setting up the correct equation \(\mathrm{900 = 5x - 220}\), leading to confusion and random guessing.
Second Most Common Error:
Poor SIMPLIFY execution: Even when students correctly set up \(\mathrm{900 = 5x - 220}\), they may make arithmetic errors. Common mistakes include calculating \(\mathrm{900 + 220}\) incorrectly or dividing 1120 by 5 incorrectly. These calculation errors lead to wrong numerical answers, even when the approach is sound.
The Bottom Line:
This problem tests whether students understand the relationship between inputs and outputs in function notation, and then requires careful algebraic manipulation. The key insight is recognizing that "earning $900 profit" means setting the profit function equal to 900, not plugging in a known number of posters.