Combined Work Rate: Pipe and Hose Filling a Pool

What This Problem Teaches

• Converting individual completion times into work rates using reciprocals
• Understanding that combined work rates add together when sources operate simultaneously
• Adding fractions with different denominators in practical applications
• Converting combined rates back to completion times
• Recognizing the mathematical principle that faster combined work emerges from rate addition, not time averaging

Solution: Method 1 — The Rate Addition Approach

The key insight is that when multiple sources work together, their rates of work add up. We need to think in terms of "pools per hour" rather than "hours per pool."

Step 1 — Convert times to work rates

Each source's work rate is the reciprocal of its completion time:

Step 2 — Add the rates to find combined rate

When both sources work together, their outputs combine:

To add these fractions, we need a common denominator. The LCM of 11 and 20 is 220:

Step 3 — Convert combined rate back to completion time

To find how long it takes to fill 1 pool at this combined rate:

Step 4 — Simplify the answer

Convert the improper fraction to a mixed number:

Converting to hours and minutes: 3/31 × 60 ≈ 5.8 minutes

Solution: Method 2 — The Harmonic Mean Formula

For exactly two sources working together, there's a direct formula that saves steps.

Step 1 — Apply the combined work formula

When two sources with times t₁ and t₂ work together, the combined time T satisfies:

Step 2 — Substitute the given times

Step 3 — Solve for T

Take the reciprocal of both sides:

This gives us the same answer: approximately 7 hours and 6 minutes.

Verification

Let's check our answer by seeing how much of the pool gets filled in 220/31 hours:

Perfect! The contributions add up to exactly one complete pool.

Common Pitfalls

The Pattern Behind This

This problem exemplifies the general principle of rate addition in parallel systems. The formula for any number of sources working together is:

Where each individual rate equals 1/(completion time).

This same mathematical structure appears in physics (electrical conductors in parallel), computer science (parallel processing), and biology (metabolic pathways). The key insight is that when systems work independently toward the same goal, their capabilities add directly.

How to Spot This Problem Type

Watch for these telltale phrases that signal a combined work rate problem:

• "If X alone takes [time], and Y alone takes [time]..."
• "Working together, how long would it take?"
• "Both pipes/hoses/machines operating simultaneously"
• Any scenario involving parallel completion of a single task

The structure is always the same: individual completion times given, combined completion time unknown. The solution path is always through rate addition, never time averaging.