Work Rate: Two Faucets Filling a Tank Together

What You Will Learn

• Converting completion times into work rates (reciprocals)
• Adding work rates when multiple workers operate simultaneously
• Converting combined rates back to completion times
• Why averaging times gives the wrong answer (and what to do instead)
• Setting up and solving fractional equations with common denominators

Solution: Method 1 — The Rate Addition Approach

The key insight is that work rates add together, not work times. When faucets work simultaneously, their individual rates of filling combine.

Step 1 — Find each faucet's rate

Convert the completion times to rates in "tanks per hour":

Step 2 — Add the rates together

When both faucets work together, their rates combine:

To add these fractions, find a common denominator of 6:

Step 3 — Convert the combined rate to time

To find the time needed, take the reciprocal of the rate:

Step 4 — Convert to hours and minutes

Convert the decimal to a more readable format:

Solution: Method 2 — The Common Work Approach

Instead of working with fractions, we can imagine a larger amount of work that makes the arithmetic cleaner.

Step 1 — Choose a convenient total amount of work

Let's say the job is filling 6 tanks instead of 1 tank. Why 6? Because 6 is the least common multiple of 2 and 3, making our rates come out to whole numbers.

Step 2 — Find each faucet's rate in tanks per hour

Scale up the rates proportionally:

Step 3 — Find the combined rate and time

Working together, they fill 3 + 2 = 5 tanks per hour:

Step 4 — Scale back to the original problem

Since we want to fill just 1 tank, not 6 tanks:

Wait, that's not right! Let me recalculate. If they take 1.2 hours to fill 6 tanks together, then for 1 tank:

Actually, let me restart this calculation properly:

Verification

Let's verify our answer by checking how much of the tank each faucet fills in 1.2 hours:

Perfect! Together they fill exactly one full tank in 1.2 hours.

We can also verify using the general formula. For work rate problems, if individual times are a and b, the combined time is:

Watch Out For These

The General Pattern

This problem follows the universal pattern for combined work rates. When workers operate simultaneously, their rates are additive:

For the special case of just two workers, this simplifies to the harmonic mean formula:

In our problem: (2 × 3) ÷ (2 + 3) = 6 ÷ 5 = 1.2 hours.

Real Applications

This exact calculation appears throughout engineering and operations:

Manufacturing: Two assembly lines producing the same product — Line A completes 50 units/hour, Line B completes 30 units/hour, together they produce 80 units/hour.

Computer networks: Parallel data streams with different bandwidths combine their rates when downloading the same file from multiple sources.

Pharmacology: When the body eliminates a drug through multiple pathways (kidney and liver), the clearance rates add to give total elimination rate.

How to Spot This Problem Type

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

• "...working together..." or "...both at the same time..."
• "How long will it take if..." followed by multiple workers
• Individual completion times given, combined time unknown
• Jobs involving filling, emptying, building, or producing

Variation alert: Sometimes the problem involves one thing helping and another thing hindering (like a faucet filling while a drain empties). In that case, subtract the hindrance rate from the help rate.

Red flag phrases: "Taking turns" or "one after the other" signal sequential work, not combined work — those are different problem types entirely.

What If?