A dance teacher ordered outfits for students for a dance recital. Outfits for boys cost $26, and outfits for girls...
GMAT Algebra : (Alg) Questions
A dance teacher ordered outfits for students for a dance recital. Outfits for boys cost \(\$26\), and outfits for girls cost \(\$35\). The dance teacher ordered a total of \(28\) outfits and spent \(\$881\). If \(\mathrm{b}\) represents the number of outfits the dance teacher ordered for boys and \(\mathrm{g}\) represents the number of outfits the dance teacher ordered for girls, which of the following systems of equations can be solved to find \(\mathrm{b}\) and \(\mathrm{g}\)?
\(26\mathrm{b} + 35\mathrm{g} = 28\)
\(\mathrm{b} + \mathrm{g} = 881\)
\(26\mathrm{b} + 35\mathrm{g} = 881\)
\(\mathrm{b} + \mathrm{g} = 28\)
\(26\mathrm{g} + 35\mathrm{b} = 28\)
\(\mathrm{b} + \mathrm{g} = 881\)
\(26\mathrm{g} + 35\mathrm{b} = 881\)
\(\mathrm{b} + \mathrm{g} = 28\)
1. TRANSLATE the problem information
- Given information:
- \(\mathrm{b}\) = number of outfits for boys
- \(\mathrm{g}\) = number of outfits for girls
- Boy outfits cost $26 each
- Girl outfits cost $35 each
- Total outfits ordered = 28
- Total amount spent = $881
2. INFER what equations we need
- We need two equations since we have two unknowns (\(\mathrm{b}\) and \(\mathrm{g}\))
- One equation for the total count constraint
- One equation for the total cost constraint
3. TRANSLATE each constraint into an equation
Total outfits constraint:
- \(\mathrm{b + g = 28}\) (boys plus girls equals total outfits)
Total cost constraint:
- Cost for boys: \(\mathrm{\$26 \times b = 26b}\) dollars
- Cost for girls: \(\mathrm{\$35 \times g = 35g}\) dollars
- Total cost: \(\mathrm{26b + 35g = 881}\) dollars
4. INFER the complete system
- Our system is:
- \(\mathrm{26b + 35g = 881}\) (total cost)
- \(\mathrm{b + g = 28}\) (total outfits)
This matches choice B.
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students mix up which numbers go with which constraints, putting the total number of outfits (28) where the total cost (881) should be, and vice versa.
Their reasoning: "I see 28 and 881 in the problem, so one goes in each equation somewhere." They don't carefully track which constraint each number represents.
This may lead them to select Choice A (\(\mathrm{26b + 35g = 28}\), \(\mathrm{b + g = 881}\)).
Second Most Common Error:
Poor TRANSLATE reasoning: Students confuse which outfit type gets which cost, writing \(\mathrm{26g + 35b}\) instead of \(\mathrm{26b + 35g}\).
Their reasoning: "I know the variables and costs go together, but I mix up which variable represents boys vs. girls."
This may lead them to select Choice C (\(\mathrm{26g + 35b = 28}\), \(\mathrm{b + g = 881}\)) or Choice D (\(\mathrm{26g + 35b = 881}\), \(\mathrm{b + g = 28}\)).
The Bottom Line:
This problem tests your ability to carefully track what each number in the word problem represents and translate it into the correct mathematical relationship. The key is methodically identifying each constraint and which numbers belong with which variables.
\(26\mathrm{b} + 35\mathrm{g} = 28\)
\(\mathrm{b} + \mathrm{g} = 881\)
\(26\mathrm{b} + 35\mathrm{g} = 881\)
\(\mathrm{b} + \mathrm{g} = 28\)
\(26\mathrm{g} + 35\mathrm{b} = 28\)
\(\mathrm{b} + \mathrm{g} = 881\)
\(26\mathrm{g} + 35\mathrm{b} = 881\)
\(\mathrm{b} + \mathrm{g} = 28\)