KA-0044Combined work rates
4 points, difficulty 2 of 3Level 5–6about 90sOne tap fills a tank in 6 hours and another fills it in 3 hours. With both running, how long does the tank take to fill?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Ask how much each tap fills in one hour.
2The strategy
Work in tanks per hour rather than hours per tank. Add the two hourly amounts to get how much the pair fills in an hour, then turn that back into a time.
3The full solution
In one hour the first tap fills one sixth and the second fills one third, which is two sixths. Together that is three sixths, or half the tank, so the whole tank takes 2 hours.
Solution
The reliable way
Switch to rates. In one hour the first tap fills 1/6 of the tank and the second fills 1/3 = 2/6. Together they fill 3/6 = 1/2 of the tank per hour, so the full tank takes 2 hours. Sanity check: two taps must be faster than the faster tap alone, and 2 hours is less than 3. The transferable idea: add rates, never times.
The elegant way
The second tap is twice as fast as the first, so together they work at three times the first tap's rate and take a third of its 6 hours.
Why this is on the test: Adding times instead of rates produces an answer slower than one tap alone, and spotting that absurdity is itself the lesson.