A hardware store sells two sizes of paint cans: small and large. The store's sales of these paint cans totaled...
GMAT Algebra : (Alg) Questions
A hardware store sells two sizes of paint cans: small and large. The store's sales of these paint cans totaled $1,482.75 last month. The equation \(7.95(\mathrm{x} + \mathrm{y}) + 4.05\mathrm{y} = 1482.75\) represents this situation, where \(\mathrm{x}\) is the number of small cans sold and \(\mathrm{y}\) is the number of large cans sold. According to the equation, what is the price, in dollars, of each small can?
Enter your answer as a decimal or as a fraction in lowest terms.
1. TRANSLATE the equation structure
- Given equation: \(7.95(\mathrm{x} + \mathrm{y}) + 4.05\mathrm{y} = 1482.75\)
- Where \(\mathrm{x}\) = number of small cans, \(\mathrm{y}\) = number of large cans
- This equation represents the total sales revenue
2. SIMPLIFY by expanding the equation
- Distribute 7.95 to both terms in parentheses:
\(7.95(\mathrm{x} + \mathrm{y}) + 4.05\mathrm{y} = 7.95\mathrm{x} + 7.95\mathrm{y} + 4.05\mathrm{y}\) - Combine like terms:
\(7.95\mathrm{x} + 7.95\mathrm{y} + 4.05\mathrm{y} = 7.95\mathrm{x} + 12.00\mathrm{y} = 1482.75\)
3. INFER the pricing information from coefficients
- In the expanded form \(7.95\mathrm{x} + 12.00\mathrm{y} = 1482.75\):
- Coefficient of x (7.95) = price per small can
- Coefficient of y (12.00) = price per large can
- Therefore, each small can costs $7.95
4. Convert to fraction form (if requested)
- \(7.95 = \frac{795}{100}\)
- Simplify by dividing both numerator and denominator by 5: \(\frac{159}{20}\)
Answer: 7.95 or 159/20
Why Students Usually Falter on This Problem
Most Common Error Path:
Poor TRANSLATE reasoning: Students may think they need to solve for specific values of x and y to determine the price, not recognizing that the equation structure directly reveals the pricing.
They might set up systems of equations or try to find specific values for the number of cans sold, leading to unnecessary complexity and confusion. This leads to abandoning systematic solution and guessing.
Second Most Common Error:
Weak INFER skill: Students may expand the equation correctly but fail to recognize that coefficients represent unit prices in revenue equations.
They might see \(7.95\mathrm{x} + 12.00\mathrm{y} = 1482.75\) but not understand that 7.95 is the price per small can. This may lead them to think additional calculations are needed, causing confusion and random answer selection.
The Bottom Line:
This problem tests whether students can interpret algebraic expressions in context rather than just manipulate them. The key insight is recognizing that in revenue equations, coefficients directly represent prices—no further solving required.