Combined Work Rate: Two Drainage Pipes

Skills This Problem Builds

• Converting individual completion times to work rates
• Understanding that combined rates add when working toward the same goal
• Using dimensional analysis to track units throughout calculations
• Recognizing the relationship between rate, time, and total work
• Applying the work rate formula to real-world drainage scenarios

Picture This

Solution: Method 1 — The Unit Rate Approach

The key insight is to convert each pipe's completion time into a drainage rate, then add those rates together.

Step 1 — Find each pipe's drainage rate

Rate equals work divided by time, so we divide the pool volume by each pipe's individual completion time.

Step 2 — Add the rates to find combined drainage

When both pipes work together, their rates add because they're both removing water from the same pool.

Step 3 — Calculate time using the combined rate

Now we divide the total work by the combined rate to find the time needed.

Solution: Method 2 — The Reciprocal Formula

There's an elegant shortcut when you're given individual completion times and need the combined time.

Step 1 — Apply the harmonic mean formula

For two workers (or pipes) that complete a job in times a and b, the combined time is ab/(a + b).

Step 2 — Understand why this works

This formula comes from adding the work fractions. Pipe 1 completes 1/180 of the job per minute, Pipe 2 completes 1/120 per minute. Together they complete (1/180 + 1/120) = 1/72 of the job per minute.

Verification

Let's verify by calculating how much work each pipe accomplishes in 72 minutes:

Watch Out For These

Recognizing This Problem Type

Look for these telltale signs that signal a combined work rate problem:

• "Working together" or "both pipes draining" — multiple agents doing the same task
• Individual completion times given — "Pipe A alone takes X minutes, Pipe B alone takes Y minutes"
• Same type of work — filling, draining, building, or completing identical tasks
• Question asks for combined time — "How long together?" or "How long with both?"

This problem structure appears in manufacturing (two machines making widgets), construction (two crews building), data processing (two servers handling requests), and anywhere multiple agents work in parallel toward the same goal.

The General Formula

For any number of workers or pipes completing the same job:

The reciprocal formula only works cleanly for two workers. With three or more, it's usually faster to convert to rates, add them, then find the time. Also remember this formula assumes all workers start and stop together—it doesn't handle scenarios where one worker leaves early or joins late.

Real Applications

Network engineering: When two internet connections are bonded, their bandwidths add. A 50 Mbps line and a 30 Mbps line together provide 80 Mbps of total throughput.

Manufacturing: Two assembly machines producing identical parts work at combined rates. If Machine A produces 100 units/hour and Machine B produces 150 units/hour, together they produce 250 units/hour.

Medical infusion: Multiple IV drips delivering the same medication combine their flow rates. This calculation ensures patients receive the intended dosage rate when multiple access points are used.