An individual is comparing the charges of two electricians. The first electrician charges a set fee for coming to the...

Phase 1: Owning the Dataset

Visual Representation

Let's create a comparison table to track the charges:

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.