A certain corporation has software that calculates the cost of a meeting. The software multiplies the hourly salary for each...
GMAT Two Part Analysis : (TPA) Questions
A certain corporation has software that calculates the cost of a meeting. The software multiplies the hourly salary for each employee attending the meeting by the number of hours that the employee spends at the meeting. The cost of the meeting is the sum of those results. One manager has used another method to calculate the cost of a meeting. The manager takes the average (arithmetic mean) salary of all employees attending the meeting multiplied by the average number of hours that employees spend at the meeting, which is then multiplied by the total number of employees attending the meeting.
Consider the following incomplete statement: The manager's calculation is equal to that of the software when all the employees who attend the meeting 1 at the meeting, or if that is not the case, when all of those employees who attend the meeting 2 Based on the information provided, select for 1 and for 2 the options that create the statement that is most accurate and rhetorically well-constructed. Make only two selections, one in each column.
Phase 1: Owning the Dataset
Understanding the Two Methods
Let's create a comparison table to understand how each method works:
| Method | Calculation |
| Software | Sum of (Each employee's salary × Their hours) |
| Manager | (Average salary) × (Average hours) × (Number of employees) |
Concrete Example
Let's test with 3 employees:
- Employee A: $50/hour, 2 hours
- Employee B: $100/hour, 1 hour
- Employee C: $75/hour, 3 hours
Software calculation:
\(($50 \times 2) + ($100 \times 1) + ($75 \times 3) = $100 + $100 + $225 = $425\)
Manager calculation:
- Average salary: \(($50 + $100 + $75) \div 3 = $75/\mathrm{hour}\)
- Average hours: \((2 + 1 + 3) \div 3 = 2 \mathrm{hours}\)
- Total: \($75 \times 2 \times 3 = $450\)
These are different! So when are they equal?
Phase 2: Understanding the Question
Mathematical Analysis
Let's express both methods algebraically:
- Software: \(\Sigma(\mathrm{salary}_i \times \mathrm{hours}_i)\)
- Manager: \((\Sigma\mathrm{salary}_i/n) \times (\Sigma\mathrm{hours}_i/n) \times n = (\Sigma\mathrm{salary}_i \times \Sigma\mathrm{hours}_i)/n\)
For these to be equal:
\(\Sigma(\mathrm{salary}_i \times \mathrm{hours}_i) = (\Sigma\mathrm{salary}_i \times \Sigma\mathrm{hours}_i)/n\)
Key Insight
This equality holds in specific cases. Let's test two scenarios:
Scenario 1: All employees spend the same time (t hours)
- Software: \(\Sigma(\mathrm{salary}_i \times t) = t \times \Sigma\mathrm{salary}_i\)
- Manager: \((\Sigma\mathrm{salary}_i/n) \times t \times n = t \times \Sigma\mathrm{salary}_i\)
- Result: Equal! ✓
Scenario 2: All employees have the same salary (s dollars/hour)
- Software: \(\Sigma(s \times \mathrm{hours}_i) = s \times \Sigma\mathrm{hours}_i\)
- Manager: \(s \times (\Sigma\mathrm{hours}_i/n) \times n = s \times \Sigma\mathrm{hours}_i\)
- Result: Equal! ✓
Phase 3: Finding the Answer
Matching to Answer Choices
We've identified two conditions that make the calculations equal:
- When all employees spend the same amount of time at the meeting
- When all employees have the same salary
Looking at our answer choices:
- "spend the same amount of time" ✓ (matches condition 1)
- "have the same salary" ✓ (matches condition 2)
- Other options don't create mathematical equality
Phase 4: Solution
Statement 1: "spend the same amount of time"
Statement 2: "have the same salary"
These selections create a mathematically accurate statement: The manager's calculation equals the software's when all employees spend the same amount of time at the meeting, or if that's not the case, when all employees have the same salary.