Combined Work Rate: When Does Dad Join?

What This Problem Teaches

• Converting between work time and work rate (the key insight that rates add, times don't)
• Setting up equations where individual contributions combine to create a known total
• Working with fractions that represent "portion of job completed per hour"
• Recognizing that faster workers do proportionally more of the work
• Understanding why averaging times gives completely wrong answers in work problems

Solution: Method 1 — The Work Rate Addition

The essential insight is that work rates add when people work together, but work times don't. We need to think in terms of "cars per hour" rather than "hours per car."

Step 1 — Convert times to work rates

Levi alone takes 2.5 hours, so his rate is 1/2.5 = 2/5 cars per hour. When they work together, they complete 1 car in 1 hour, so their combined rate is 1/1 = 1 car per hour.

Step 2 — Set up the rate equation

When working together, individual rates add up to the combined rate:

Step 3 — Solve for Dad's rate

Dad's rate = 1 - 2/5 = 5/5 - 2/5 = 3/5 cars per hour

Step 4 — Convert Dad's rate back to time

If Dad completes 3/5 of a car per hour, then to complete 1 whole car takes 1 ÷ (3/5) = 1 × (5/3) = 5/3 hours.

5/3 hours = 5/3 × 60 = 100 minutes = 1 hour and 40 minutes

Solution: Method 2 — The Fraction Visualization

Let's visualize what happens during that 1 hour when they work together:

Step 1 — Find Levi's contribution in 1 hour

In 1 hour, Levi completes 1 ÷ 2.5 = 2/5 of the car.

Step 2 — Find Dad's contribution

Since the whole car gets done in 1 hour, Dad must complete the remaining fraction: 1 - 2/5 = 3/5 of the car.

Step 3 — Scale up to find Dad's solo time

If Dad does 3/5 of the car in 1 hour, then to do the full car (5/5), he needs (5/5) ÷ (3/5) = 5/3 hours.

Verification

Let's check: Levi's rate is 2/5 cars/hour, Dad's rate is 3/5 cars/hour. Together: 2/5 + 3/5 = 5/5 = 1 car per hour. ✓

This means they finish 1 car in exactly 1 hour when working together, which matches the problem statement.

Double-check Dad's time: At 3/5 cars per hour, for 1 full car: 1 ÷ (3/5) = 5/3 = 1.67 hours = 100 minutes. ✓

Watch Out For These

The General Pattern

For any two-person work problem:

Given any two of these values, you can solve for the third. In our problem: 1/2.5 + 1/B = 1/1, so 1/B = 1 - 1/2.5 = 3/5, giving B = 5/3 hours.

Key insight: The person who works faster (smaller time) contributes a larger fraction of the combined work. This is why averaging times fails—it assumes equal contribution regardless of speed.

How to Spot This Problem Type

• "Working together" or "working simultaneously" — signals that rates combine
• "Alone" vs. "together" time comparisons — you'll need the work rate formula
• Two different completion times given — typically one person alone and both together
• Asked to find the other person's solo time — requires solving the rate equation

These problems appear in algebra, SAT/ACT prep, and real-world contexts like manufacturing, project management, and resource allocation.