A landscaping company earns a profit of $4,500 on a large project. This profit is to be distributed equally among...
GMAT Algebra : (Alg) Questions
A landscaping company earns a profit of \(\$4,500\) on a large project. This profit is to be distributed equally among the \(\mathrm{n}\) team members who worked on the project, where \(\mathrm{n}\) is a positive integer. Which function \(\mathrm{p}\) models the amount of money, in dollars, each team member will receive?
\(\mathrm{p(n) = 4500n}\)
\(\mathrm{p(n) = \frac{4500}{n}}\)
\(\mathrm{p(n) = \frac{n}{4500}}\)
\(\mathrm{p(n) = 4500 - n}\)
1. TRANSLATE the problem information
- Given information:
- Total profit to distribute: $4,500
- Number of team members: n (positive integer)
- Distribution method: 'equally among' the team members
- Need to find: function \(\mathrm{p(n)}\) for amount each member receives
2. INFER the mathematical operation needed
- 'Distributed equally' means we divide the total by the number of people
- Each person gets the same amount, so: individual share = total ÷ number of people
- This gives us: \(\mathrm{p(n) = \frac{4500}{n}}\)
3. INFER why other choices don't work
- Choice A (\(\mathrm{4500n}\)): As more people join, each person would get MORE money - this contradicts equal sharing
- Choice C (\(\mathrm{\frac{n}{4500}}\)): Each person would get less than $1 regardless of team size - this doesn't make sense
- Choice D (\(\mathrm{4500 - n}\)): This just subtracts the number of people from the total - not equal distribution
4. Verify the answer makes sense
- If n = 1: \(\mathrm{p(1) = \frac{4500}{1} = \$4,500}\) (one person gets everything)
- If n = 9: \(\mathrm{p(9) = \frac{4500}{9} = \$500}\) (nine people split equally)
- If n = 18: \(\mathrm{p(18) = \frac{4500}{18} = \$250}\) (eighteen people split equally)
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students may not recognize that 'distributed equally among n team members' specifically means division. They might think about the problem in terms of 'n team members getting money' and mistakenly choose multiplication, leading them to select Choice A (\(\mathrm{4500n}\)).
This error occurs because they focus on 'n team members' and '4500 dollars' but miss the crucial word 'equally' that indicates each person gets a fraction of the total.
The Bottom Line:
The key challenge is translating the everyday phrase 'distributed equally among' into its mathematical equivalent: division. Students must recognize that equal distribution always involves dividing a total by the number of recipients.
\(\mathrm{p(n) = 4500n}\)
\(\mathrm{p(n) = \frac{4500}{n}}\)
\(\mathrm{p(n) = \frac{n}{4500}}\)
\(\mathrm{p(n) = 4500 - n}\)