Work Rate Problems

Problem Setup

1/6 + 1/4 = 1/T

Worker A alone (hours)

Worker B alone (hours)

Enter the hours each worker takes alone. Find how long together.

Examples

1/t₂ = 1/3 − 1/5

Together T (hours)

Worker A alone (hours)

Know the combined time T and Worker A's solo time. Find Worker B's solo time.

Examples

1/6 + 1/8 − 1/12 = 1/T

Fill pipe 1 (hours)

Fill pipe 2 (hours)

Drain pipe (hours)

Two pipes fill the tank; one pipe drains it. Find the net fill time.

Examples

Solution

Enter values above and press Solve to see the time T, rate equation, and a verification check.

Time Together (T)

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Rate Equation

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Verification (rates × T = 1 job)

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Work Rate Formula

Rate × Time = Work done 1/t = fraction of job per hour Combined rate = 1/t₁ + 1/t₂

Each worker's rate is how much of the job they finish per unit of time. If Worker A takes t₁ hours alone, their rate is 1/t₁ jobs per hour.

When two people work together, their rates add. Setting the combined rate equal to 1/T and solving gives the total time T.

For the formula directly: T = (t₁ · t₂) / (t₁ + t₂). This is the harmonic-mean relationship between the two times.

To find one worker's unknown time: rearrange to 1/t₂ = 1/T − 1/t₁, then take the reciprocal.

Key Insight

Each rate is the fraction of the job done per unit time

Add rates when workers (or pipes) work together toward the same goal. Subtract rates when a drain pipe opposes the fill pipes.

The net rate determines how quickly the whole job gets done. If net rate is negative, the tank never fills — the drain is faster than the combined fill.

Always verify: multiply the net combined rate by T to confirm the result equals exactly 1 complete job.

Rate of Worker A: 1/t₁ jobs per hour.
Together: 1/t₁ + 1/t₂ = 1/T → T = t₁·t₂/(t₁+t₂).
Subtract drain: 1/t₁ + 1/t₂ − 1/t₃ = 1/T.
T is always less than the fastest worker's solo time.

Work rate problems still tricky?