KA-0159Combined work rates
5 points, difficulty 3 of 3Level 7–8about 135sOne tap fills a tank in 6 hours. A second tap fills the same tank in 4 hours. Both taps are opened together on an empty tank. How long does it take to fill?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Times do not add. Think about how much each tap does in one hour.
2The strategy
In one hour the first tap fills 1/6 of the tank and the second fills 1/4. Together that is 1/6 + 1/4 of a tank per hour.
3The full solution
1/6 + 1/4 = 5/12 of a tank per hour, so a whole tank takes 12/5 = 2.4 hours, which is 2 hours 24 minutes.
Solution
The reliable way
Work in rates rather than times, because rates add and times do not. In one hour the first tap fills 1/6 of the tank and the second fills 1/4, so together they fill 1/6 + 1/4 = 2/12 + 3/12 = 5/12 of the tank each hour. Filling the whole tank therefore takes 1 divided by 5/12 = 12/5 = 2.4 hours, which is 2 hours and 24 minutes. Sanity check: the answer must be less than 4 hours, the faster tap's time on its own. The transferable idea: rates add, times do not.
The elegant way
In 12 hours the taps would fill 2 and 3 tanks, so 5 tanks in 12 hours, which is 12/5 hours per tank.
Why this is on the test: Adding times instead of rates is the standard trap, and the sanity check - faster than either alone - catches it every time.