A factory makes 9-inch, 7-inch, and 4-inch concrete screws. During a certain day, the number of 9-inch concrete screws that...
GMAT Algebra : (Alg) Questions
A factory makes 9-inch, 7-inch, and 4-inch concrete screws. During a certain day, the number of 9-inch concrete screws that the factory makes is 5 times the number \(\mathrm{n}\) of 7-inch concrete screws, and the number of 4-inch concrete screws is \(22\). During this day, the factory makes \(100\) concrete screws total. Which equation represents this situation?
\(9(5\mathrm{n}) + 7\mathrm{n} + 4(22) = 100\)
\(9\mathrm{n} + 7\mathrm{n} + 4\mathrm{n} = 100\)
\(5\mathrm{n} + 22 = 100\)
\(6\mathrm{n} + 22 = 100\)
1. TRANSLATE the problem information
- Given information:
- Number of 7-inch screws: \(\mathrm{n}\)
- Number of 9-inch screws: "5 times the number n of 7-inch screws" → \(\mathrm{5n}\)
- Number of 4-inch screws: \(22\)
- Total number of screws: \(100\)
2. INFER what the equation should represent
- We need an equation for the total NUMBER of screws, not their total length
- The key insight: We're counting screws, not measuring inches
3. TRANSLATE the relationship into an equation
- Total screws = (9-inch screws) + (7-inch screws) + (4-inch screws)
- \(\mathrm{100 = 5n + n + 22}\)
4. SIMPLIFY by combining like terms
- \(\mathrm{5n + n = 6n}\)
- So our equation becomes: \(\mathrm{6n + 22 = 100}\)
Answer: D. \(\mathrm{6n + 22 = 100}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE reasoning: Students confuse what the equation should represent and mix up the lengths of the screws with the number of screws.
They might think the 9-inch, 7-inch, and 4-inch measurements should be part of the equation, leading them to create something like \(\mathrm{9(5n) + 7n + 4(22) = 100}\). This would represent the total length in inches of all screws being 100, not the total number of screws being 100.
This leads them to select Choice A (\(\mathrm{9(5n) + 7n + 4(22) = 100}\)).
Second Most Common Error:
Incomplete TRANSLATE execution: Students correctly identify that they need to count screws but miss part of the relationship.
They might forget to include the 7-inch screws (represented by \(\mathrm{n}\)) in their equation, thinking only about the 9-inch screws (\(\mathrm{5n}\)) and 4-inch screws (\(22\)). This creates the equation \(\mathrm{5n + 22 = 100}\).
This may lead them to select Choice C (\(\mathrm{5n + 22 = 100}\)).
The Bottom Line:
This problem tests whether students can distinguish between counting objects versus measuring their attributes, and whether they can accurately translate all the given relationships into a single equation.
\(9(5\mathrm{n}) + 7\mathrm{n} + 4(22) = 100\)
\(9\mathrm{n} + 7\mathrm{n} + 4\mathrm{n} = 100\)
\(5\mathrm{n} + 22 = 100\)
\(6\mathrm{n} + 22 = 100\)