Work Rate Problem: Kendall and Dale Cleaning Together

What You Will Learn

• How to convert individual completion times into work rates (jobs per hour)
• Why work rates add when people work together, but completion times don't
• The relationship between rate and time: Time = Work ÷ Rate
• How to add fractions with different denominators in a practical context
• Recognition of the classic work rate problem structure that appears across mathematics

Solution: Method 1 — The Rate Addition Approach

The key insight is that when people work together, their rates add up, not their times. Let's think about how much of the job each person completes per hour.

Step 1 — Find each person's work rate

Kendall completes 1 garage in 6 hours, so his rate is 1/6 garage per hour.

Dale completes 1 garage in 9 hours, so his rate is 1/9 garage per hour.

Step 2 — Add their rates to find the combined rate

When working together, their combined rate is:

To add these fractions, we need a common denominator. The LCD of 6 and 9 is 18:

1/6 = 3/18 and 1/9 = 2/18

So: 1/6 + 1/9 = 3/18 + 2/18 = 5/18 garage per hour

Step 3 — Calculate the time to complete one garage

Using the formula Time = Work ÷ Rate:

18/5 = 3.6 hours

Step 4 — Convert to hours and minutes

The decimal part: 0.6 hours × 60 minutes/hour = 36 minutes

So 3.6 hours = 3 hours and 36 minutes

Solution: Method 2 — The Shortcut Formula

There's a neat formula for two-worker problems that comes from the algebra we just did. For two people with individual times a and b, their combined time is:

Step 1 — Apply the formula

Here, a = 6 hours and b = 9 hours:

T = (6 × 9) ÷ (6 + 9) = 54 ÷ 15 = 18/5 = 3.6 hours

Step 2 — Where does this formula come from?

The combined rate is 1/a + 1/b = (b + a)/(ab)

The time is the reciprocal of the rate: Time = 1 ÷ [(a + b)/(ab)] = ab/(a + b)

This confirms our answer and gives us a handy shortcut for future problems.

Verification

Let's check our answer by seeing how much work gets done in 3.6 hours:

Kendall's contribution: 3.6 hours × (1/6 garage per hour) = 3.6/6 = 0.6 garage

Dale's contribution: 3.6 hours × (1/9 garage per hour) = 3.6/9 = 0.4 garage

Total work: 0.6 + 0.4 = 1.0 garage ✓

Perfect! The fractions add up to exactly one complete garage.

Common Pitfalls

The Pattern Behind This

Work rate problems follow a consistent pattern that appears throughout mathematics:

For n workers with times t₁, t₂, ..., tₙ, the combined time is:

This is the harmonic mean of the individual times. Notice how it's always less than the arithmetic mean (simple average) and always less than the fastest individual time — which is exactly what we expect when people collaborate.

Real Applications

Network Computing: When multiple processors work on the same computational task, their processing rates add up just like work rates. A 2 GHz processor and a 3 GHz processor together don't average to 2.5 GHz — they provide 5 GHz of combined processing power.

Manufacturing: Production lines with multiple workers or machines follow the same mathematics. If Machine A produces 100 units per hour and Machine B produces 150 units per hour, together they produce 250 units per hour, not some average.

Medical Clearance: Drug clearance from the body involves multiple pathways (kidney, liver, etc.) that work in parallel. The total clearance rate is the sum of the individual pathway rates.

What If?