How Long to Rinse Dishes Working Together?

What This Problem Teaches

• Converting individual completion times into work rates
• Understanding that work rates add when people work together
• Recognizing why averaging times gives the wrong answer
• Working with unit fractions and finding common denominators
• Connecting rate calculations to real-world efficiency problems

Solution: Method 1 — The Work Rate Approach

The key insight is that when people work together, their rates of work add up, not their times. Let's find each person's work rate first.

Step 1 — Find Chris's work rate

Chris completes the entire job in 72 minutes, so his rate is:

Step 2 — Find Bill's work rate

Bill completes the entire job in 144 minutes, so his rate is:

Step 3 — Add their work rates together

When working together, their combined rate is the sum of their individual rates:

Step 4 — Calculate the time to complete the job

If they complete 1/48 of the job per minute, then the time to complete the entire job is:

Solution: Method 2 — The Dishes-Per-Minute Approach

Instead of thinking about work rates as fractions of the job, we can calculate how many dishes each person rinses per minute.

Step 1 — Find Chris's speed

Chris rinses 720 dishes in 72 minutes:

Step 2 — Find Bill's speed

Bill rinses 720 dishes in 144 minutes:

Step 3 — Calculate their combined speed

Working together, they rinse:

Step 4 — Find the time to rinse all dishes

To rinse 720 dishes at 15 dishes per minute:

Verification

Let's check our answer by calculating how much work each person completes in 48 minutes:

We can also verify using the dishes-per-minute approach:

Common Pitfalls

✗ Mistake 1: Averaging the times

✗ Mistake 2: Adding the rates incorrectly

✗ Mistake 3: Using the harmonic mean formula incorrectly

The General Pattern

For any combined work problem, the universal approach is:

For our problem: t = (72 × 144) ÷ (72 + 144) = 10,368 ÷ 216 = 48 minutes.

This formula works because work rates are additive — when people collaborate, their individual rates of progress combine. The same principle applies to water flowing through multiple pipes, data processing on multiple servers, or any scenario where independent agents contribute simultaneously to a shared task.

Where This Shows Up in Real Life

Combined work rate calculations appear frequently in:

• Manufacturing and Production: Assembly lines where multiple workers or machines contribute to output, helping optimize staffing levels and predict completion times.
• Computer Science: Parallel processing where multiple CPUs work on the same computational task, or network bandwidth calculations when data flows through multiple channels.
• Project Management: Determining realistic timelines when multiple team members with different skill levels tackle portions of a project simultaneously.

What If?