Math PicnicFree printable practice

Work and pipes word problems worksheets

One person paints a room in six hours, another in twelve. Together? Not eighteen, and not nine. These free printable word problems build the one idea that makes work problems tractable: add the rates, not the times.

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Work and pipes word problems
Work and pipes word problems worksheet — Two workers, one job, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

The mistake everyone makes first is to average or add the times, and the fix is to think in rates instead. A painter who takes six hours does one sixth of the room an hour; the other does one twelfth; together they do a quarter of the room an hour, so the room takes four hours. Turning each time into a per-hour rate is the whole method.

Why this sheet works

Pipes filling and draining a tank are the same problem with a sign. A drain works against the fill, so its rate is subtracted, and a child who has the rate idea handles that without a new rule — the tap adds, the drain takes away, and the net rate gives the time.

How this one works

One question from this sheet, worked through a step at a time.

How long will it take to fill the tank?

A tank has two faucets. One fills it alone in 5 hours, the other in 12 hours. Both faucets are opened together.
1/5 + 1/12
1/5 + 1/12 = 17/60
1/5 + 1/12 = 17/60
1 ÷ 17/60 = 3 9/17 hours
asked:How long will it take to fill the tank?
3 9/17 hours
  1. Read it through once. Two pipes fill the SAME tank. The trick is that SPEEDS of work add even when times do not — two pipes do not fill it in the sum of their two times.
  2. Turn each pipe into how much it does in ONE hour: a pipe that fills the tank in n hours does 1/n of it each hour. Here that is 1/5 + 1/12.
  3. Working together, the fractions per hour ADD: 1/5 + 1/12 = 17/60 of the tank each hour.
  4. The two pipes fill 17/60 of the tank each hour, so the whole tank takes the RECIPROCAL of that rate — flip 17/60 over: 1 ÷ 17/60 = 3 9/17 hours.
  5. Answer the question that was asked — "How long will it take to fill the tank?" — 3 9/17 hours.

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