EQN-02-09

Combined Work & Rate

Add rates, not times, to solve "how long together" problems — the single rule that cracks work questions.

~5 min read•Difficulty 3/5

If a worker finishes a job in t units of time, the rate is 1/t of the job per unit. The whole job is "1".

combined rate = 1/ a + 1/ b

time together = 1/ (combined rate)

You add rates, never times.

A finishes in 4 hours (rate ¼), B in 6 hours (rate ⅙).

combined = 1/4 + 1/6 = 3/12 + 2/12 = 5/12 job per hour

time = 1/ (5/12) = 12/5 = 2.4 hours

The together-time is always less than the faster worker's time — a built-in sanity check.

A drain works against a fill: subtract its rate.

fill 1/3 per hour, drain 1/5 per hour \to net = 1/3 - 1/5 = 2/15 per hour

Worked examples

Example 1 (difficulty 2/5)

Pipe A fills a tank in 3 hours, pipe B in 6 hours. How long together?

◎ Predict the answer first:

Example 2 (difficulty 3/5)

A tap fills a tank in 4 hours; a leak empties it in 12 hours. With both open, how long to fill?

◎ Predict the answer first:

Convert each worker/pipe to a rate of 1/time, add (or subtract) the rates, then invert for the time.
The combined time must be smaller than the fastest single time — check it.
A drain or leak subtracts its rate from the fill rate.

Averaging the two times to get the combined time

Why: Averaging feels natural.

Fix: Add the rates

(1/ a + 1/ b),

then invert — not the times.

Adding a leak’s rate instead of subtracting it

Why: Both are "rates", so they get added.

Fix: A leak works against the fill — subtract its rate.