Work Rate Problem Solver

The Work Rate Problem Solver covers the five most common "A and B working together" word problems in one place: the classic combined-time problem, the missing-worker problem where you know one worker and the team time but not the other worker, the fill-pipe-vs-drain-pipe net-time problem, the alternating-shift problem where two workers take turns, and the partial-completion problem where one worker starts solo and the other joins midway. Type in solo times in your preferred unit — hours, minutes, days, or seconds — and the solver applies the rate-addition law, walks through the algebra step by step in LaTeX, and shows an animated dual-pie visualization with one slice per worker that grows in proportion to that worker's rate.

How to use this solver

1. Pick the scenario that matches your problem from the dropdown — together, missing worker, pipe fill vs drain, alternating shifts, or A solo then B joins.
2. Pick the time unit (hours, minutes, days, or seconds). All inputs use the same unit.
3. Enter each worker's solo time. For the missing-worker case, also enter the together time. For pipes, enter fill and drain times. For alternating shifts, enter the shift length and who starts. For partial completion, enter how long A works alone before B joins.
4. Click Solve. The headline value is the missing quantity — combined time, B's solo time, net fill time, total elapsed time, or total project time.
5. Watch the dual-pie chart fill in proportion to each worker's rate, and read the LaTeX-formatted step-by-step explanation.

The five formulas at a glance

The rate-addition law (the key idea)

Every work-rate problem reduces to one identity: rates add when workers cooperate, but times do not. If A finishes one whole job in \( T_A \), then A does \( 1/T_A \) of the job per unit time. Two workers contribute their per-unit-time fractions independently:

\[ \frac{1}{T} \;=\; \frac{1}{T_A} + \frac{1}{T_B} \]

Each scenario in this calculator is just a different unknown in this same equation:

• Together — solve for \( T \) given \( T_A \) and \( T_B \).
• Missing — solve for \( T_B \) given \( T_A \) and the together time \( T \).
• Pipe — flip the sign of one term: \( 1/T = 1/T_f - 1/T_d \).
• Alternate — break time into A+B cycles, each cycle does \( L(r_A+r_B) \) of the job.
• Partial — split the timeline: A solo, then together.

Worked example: two painters

Painter A can finish a wall in 6 hours. Painter B can do the same wall in 4 hours. How long do they take working together?

1. A's rate: \( r_A = 1/6 \) wall per hour.
2. B's rate: \( r_B = 1/4 \) wall per hour.
3. Combined rate: \( r_A + r_B = 1/6 + 1/4 = 2/12 + 3/12 = 5/12 \) wall per hour.
4. Combined time: \( T = 1 / (5/12) = 12/5 = 2.4 \) hours = 2 hours 24 minutes.
5. Notice: the answer (2.4 h) is less than both 4 h and 6 h — adding a second worker can only speed up the job.

Worked example: missing helper

You know that A alone takes 5 hours. With an unknown helper B, the team finishes in 2 hours. How long would B take alone?

1. Combined rate: \( r_T = 1/2 = 0.5 \) job per hour.
2. A's rate: \( r_A = 1/5 = 0.2 \) job per hour.
3. Subtract: \( r_B = 0.5 - 0.2 = 0.3 \) job per hour.
4. B's solo time: \( T_B = 1/0.3 \approx 3.33 \) hours.

Worked example: fill pipe vs drain pipe

A fill pipe fills a tank in 5 hours. A drain pipe empties the same tank in 8 hours. Both are open. How long until the tank is full?

1. Fill rate: \( r_f = 1/5 = 0.20 \) tank per hour.
2. Drain rate: \( r_d = 1/8 = 0.125 \) tank per hour.
3. Net rate: \( r_{net} = 0.20 - 0.125 = 0.075 \) tank per hour.
4. Time to fill: \( T = 1/0.075 \approx 13.33 \) hours = 13 h 20 min.
5. Sanity check: the fill alone is 5 h; with the drain working against it, the time more than doubles. If the drain were faster than the fill, the tank would never fill.

Worked example: alternating one-hour shifts

A finishes a job alone in 6 hours. B alone takes 8 hours. They take turns in one-hour shifts, A first. How long does the job take?

1. Per-pair work: \( L(r_A + r_B) = 1 \cdot (1/6 + 1/8) = 7/24 \approx 0.2917 \) of the job per pair.
2. Three full pairs (6 hours) finish \( 3 \cdot 7/24 = 21/24 = 0.875 \) of the job.
3. Remaining: 0.125. A's next 1-hour shift finishes \( 1/6 \approx 0.1667 \), which is more than 0.125, so A finishes during their 4th shift.
4. Time A needs to do the last 0.125: \( 0.125 / (1/6) = 0.75 \) hour.
5. Total time: \( 6 + 0.75 = 6.75 \) hours = 6 h 45 min.

Worked example: partial completion

A starts a job alone (A solo time 6 h). After 2 hours B joins (B solo time 4 h). How long until done?

1. Work A finishes alone: \( (1/6) \cdot 2 = 1/3 \) of the job.
2. Remaining work: \( 1 - 1/3 = 2/3 \).
3. Combined rate: \( 1/6 + 1/4 = 5/12 \) per hour.
4. Time for the together phase: \( (2/3) / (5/12) = (2/3) \cdot (12/5) = 8/5 = 1.6 \) hours.
5. Total: \( 2 + 1.6 = 3.6 \) hours = 3 h 36 min.

Common pitfalls and how to avoid them

• Adding times instead of rates — the most common student mistake. If A takes 6 h and B takes 4 h, the answer is NOT 5 h (the average) and NOT 10 h (the sum). It is 2.4 h, found by adding the rates.
• Together time shorter than the faster worker — together time must be less than both \( T_A \) and \( T_B \). If you computed something larger, you made an arithmetic error.
• Pipes that never fill — if the drain rate is greater than or equal to the fill rate, no amount of time will fill the tank. The solver will warn you.
• Mixing time units — converting some inputs to minutes and others to hours produces nonsense. Pick one unit at the top and use it everywhere.
• Alternating shifts: don't forget the partial last shift — after counting full A+B cycles, the work usually finishes mid-shift. Solve for that final partial shift exactly.
• Partial completion: A may finish before B joins — if \( r_A \cdot t_{solo} \geq 1 \), A is already done and B never works. The calculator handles this case automatically.

Quick reference — rate vs time

Where work-rate problems show up in real life

• Construction and contracting — estimating how long a crew of two will take when each member's solo pace is known.
• Pipes and plumbing — sizing a pump and an overflow drain so the tank reaches a target level in a set time.
• Software and CI — two test runners executing in parallel; the wall-clock equals the slowest one's time, but throughput equals the sum of rates.
• Manufacturing — multiple machines on the same line; total throughput is the sum of per-machine throughput.
• Education — work-rate problems are a staple of SAT/ACT, GRE, GMAT, and most algebra textbooks (chapter on rational equations).

Frequently asked questions