KA-0116Combined work rates
4 points, difficulty 2 of 3Level 5–6about 90sOne pipe fills a pool in 12 hours and another fills it in 4 hours. With both running, how many hours does the pool take to fill?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
How much does each pipe fill in an hour?
2The strategy
Work in pools per hour rather than hours per pool. Add the two hourly amounts to get the pair's rate, then turn that back into a time.
3The full solution
In an hour the pipes fill one twelfth and one quarter, which is one twelfth plus three twelfths = four twelfths, or a third. So 3 hours.
Solution
The reliable way
Switch to rates. In one hour the first pipe fills 1/12 of the pool and the second fills 1/4 = 3/12. Together they fill 4/12 = 1/3 of the pool per hour, so the whole pool takes 3 hours. Sanity check: two pipes must beat the faster pipe alone, and 3 is less than 4. The transferable idea: add rates, never times - and check the answer beats the faster worker on its own.
The elegant way
The second pipe is three times as fast as the first, so together they work at four times the first pipe's rate: 12 divided by 4 = 3 hours.
Why this is on the test: Adding the times gives an answer slower than one pipe alone, and noticing that absurdity is itself the lesson.