“A finishes in 3 hours, B in 6 — together?” The trap is averaging the times. understand → set up the rates → solve → verify the shares add to one job. Pick the story shape and watch the rate equation appear.
Each worker’s solo time is known. Working at the same time, how fast do they finish one job?
In short
Work problems run on rates, not times: a worker who finishes a job in 3 hours does one third of it per hour. Rates add when people work together, so the setup is combined rate × time = one whole job. VerifySolve builds that equation in stages and verifies the answer by checking that everyone's shares add to one finished job.
A solo time is a rate in disguise: 1 ÷ hours. Everything on this page is that conversion plus one rule — rates add, and the shares must total one whole job.
One worker’s rate
Two together
Three together
Find one worker
Each note re-adds the workers’ shares from the answer time. If the shares do not total exactly 1, the setup was wrong — run them live in the solver above.
A can paint a room in 3 hours, B in 6 hours. How long together?
2 hours together
Verified: A does 2/3 of the room, B does 1/3 — and 2/3 + 1/3 = 1 whole room ✓.
Three machines finish a batch in 4, 6 and 12 hours alone. All three at once?
2 hours together
Verified: shares 1/2 + 1/3 + 1/6 = 1 whole batch ✓.
A (6 h alone) works 2 hours alone, then B (3 h alone) joins. How much longer?
4/3 hours more — 10/3 hours from the start
Verified: A’s total share 5/9 plus B’s share 4/9 = 1 whole job ✓.
Together they take 2 hours. A alone takes 3 hours. How long would B take alone?
B alone would take 6 hours
Verified: (1/3 + 1/6) × 2 = 1 — the pair really does finish in 2 hours ✓.
3 hours and 6 hours do not average to 4.5 hours together — the pair must be faster than either worker alone. Averaging times is the most common work-problem error, and the answer it gives is always too slow.
Times never add in work problems. What adds is the fraction of the job done per hour: 1/3 + 1/6 = 1/2 a job per hour, which is why the pair finishes in 2 hours. Add rates, then invert once.
When one worker starts alone, part of the job is already done before the second arrives. The combined rate applies only to what remains — applying it to the whole job double-counts the head start.
Rate × time = one job, where rate = 1 ÷ (hours alone). A worker who finishes in 3 hours does 1/3 of the job per hour. When workers team up, their rates add — so the together-time comes from the combined rate, never from averaging the times.
Because the faster worker does more of the job, so a plain average over-weights the slow worker. A 3-hour worker and a 6-hour worker finish together in 2 hours — faster than either alone, and nowhere near the 4.5-hour average. Rates, not times, are what add.
The same way: add a third rate. 1/a + 1/b + 1/c is the combined rate, and one job divided by that rate is the together-time. The verification checks that all three shares add to exactly one finished job.
First subtract what the early worker already finished — that share of the job is gone. Then apply the combined rate only to what remains. The solver splits the story exactly that way and reports both the extra time and the total time from the start.
Each worker’s share of the job is recomputed from the answer time — share = time ÷ solo time — and the shares must add to exactly 1, one whole job. A wrong setup shows up immediately: the shares land above or below 1.
Yes — free, no sign-up, no app, with the full staged solution and share verification included.
The rate family’s sibling: speeds combining the same way work rates do.
Two-equation stories: percents and prices, in stages.
The staged method explained, plus every word-problem family we cover.
Already have the rate equation? Solve it directly, with plug-back proof.
A note on honest use: VerifySolve is built for learning. Use the steps to understand the method, then close the tab and try one on your own — that is where the grade actually comes from. If your teacher or school has a policy on calculators, follow it.