An individual is comparing the charges of two electricians. The first electrician charges a set fee for coming to the...
GMAT Two Part Analysis : (TPA) Questions
An individual is comparing the charges of two electricians. The first electrician charges a set fee for coming to the individual's home and charges an hourly rate that is half of the second electrician's hourly rate. The second electrician charges no fees for coming to the individual's home. The individual will have no charges other than those mentioned.
Select for First electrician's set fee and for Second electrician's hourly rate the two figures, in US dollars ($), that could be the first electrician's set fee and the second electrician's hourly rate such that both figures would result in the two electricians charging the same amount for coming to the individual's home and working for exactly one hour. Make only two selections, one in each column.
Phase 1: Owning the Dataset
Visual Representation
Let's create a comparison table to track the charges:
| Electrician | Set Fee | Hourly Rate | Total for 1 hour |
| First | F | \(\mathrm{R2}/2\) | \(\mathrm{F} + \mathrm{R2}/2\) |
| Second | $0 | R2 | R2 |
Where:
- F = First electrician's set fee (unknown)
- R2 = Second electrician's hourly rate (unknown)
- \(\mathrm{R2}/2\) = First electrician's hourly rate (half of second's)
Phase 2: Understanding the Question
Breaking Down the Problem
We need both electricians to charge the same amount for 1 hour of work:
- First electrician's total: \(\mathrm{F} + \mathrm{R2}/2\)
- Second electrician's total: R2
Setting them equal:
\(\mathrm{F} + \mathrm{R2}/2 = \mathrm{R2}\)
Solving for the Relationship
\(\mathrm{F} = \mathrm{R2} - \mathrm{R2}/2\)
\(\mathrm{F} = \mathrm{R2}/2\)
Key Insight: The first electrician's set fee must equal exactly half of the second electrician's hourly rate.
Phase 3: Finding the Answer
Systematic Checking
Since \(\mathrm{F} = \mathrm{R2}/2\), we know that \(\mathrm{R2} = 2\mathrm{F}\). Let's check each possible set fee value:
If F = $10 → \(\mathrm{R2} = 2 \times \$10 = \$20\)
Is $20 in our choices? No, continue.
If F = $30 → \(\mathrm{R2} = 2 \times \$30 = \$60\)
Is $60 in our choices? No, continue.
If F = $50 → \(\mathrm{R2} = 2 \times \$50 = \$100\)
Is $100 in our choices? Yes! ✓
? Stop here - we found our answer.
Verification
Let's confirm our answer works:
- First electrician (1 hour): \(\$50 + (\$100/2) = \$50 + \$50 = \$100\)
- Second electrician (1 hour): $100
Both charge $100 for one hour of work. ✓
Phase 4: Solution
Final Answer:
- First electrician's set fee: $50
- Second electrician's hourly rate: $100
These values satisfy our requirement that both electricians charge the same amount ($100) for coming to the home and working for exactly one hour.