Work and pipes word problems, step by step

One person paints a room in six hours, another in twelve.

The question

How long until the tank is full?

One pipe can fill a tank in 4 hours; a second pipe fills it in 10 hours. Both pipes are opened at once.

1/4 + 1/10

1/4 + 1/10 = 7/20

1 ÷ 7/20 = 2 6/7 hours

asked:How long until the tank is full?

2 6/7 hours

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.
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 1/4 + 1/10.
Working together, the fractions per hour ADD: 1/4 + 1/10 = 7/20 of the tank each hour.
The two pipes fill 7/20 of the tank each hour, so the whole tank takes the RECIPROCAL of that rate — flip 7/20 over: 1 ÷ 7/20 = 2 6/7 hours.
Answer the question that was asked — "How long until the tank is full?" — 2 6/7 hours.

How it works.

01
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.
02
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 1/4 + 1/10.
03
Working together, the fractions per hour ADD: 1/4 + 1/10 = 7/20 of the tank each hour.
04
The two pipes fill 7/20 of the tank each hour, so the whole tank takes the RECIPROCAL of that rate — flip 7/20 over: 1 ÷ 7/20 = 2 6/7 hours.
05
Answer the question that was asked — "How long until the tank is full?" — 2 6/7 hours.