AMC 8 Daily Practice Round 8
Complete problem set with solutions and individual problem pages
In a certain place, there are two ponds, Pond and Pond . Due to planned construction work, the water in both ponds must be pumped out, even during the rainy season. Pond has an area of acres. Using water pumps, it can be emptied in days; if water pumps are used instead, it will take days to empty it. Pond has an area times that of Pond . If it is to be emptied in days, how many water pumps will be needed?
- A.
- B.
- C.
- D.
- E.
Let the original amount of water in Pond be , and let the daily rainfall into Pond be .
Then both represent the amount of water a single pump can remove from Pond in one day.
That is: which gives , so
Therefore, the amount of water one pump can remove from Pond in one day is:
From this and the above, we know that the rainfall into Pond in one day is exactly the amount one pump can remove in one day.
Since Pond ’s area is times that of Pond , removing Pond ’s daily rainfall requires pumps.
The original amount of water in Pond is .
Since one pump can remove from Pond ’s original water in one day, it would take: for one pump to remove Pond ’s original water.
If we want to remove it in days, we need pumps.
Therefore, to remove both the daily rainfall and the original water from Pond in days, we need: pumps.
Arithmetic method
(1) For Pond (including rainfall), using pumps takes days to empty it; to empty it in day requires pumps.
Similarly, using pumps takes days to empty it; to empty it in day requires pumps.
The extra days mean extra days of rainfall, which requires extra pumps.
Therefore, the daily rainfall into Pond requires pump to remove.
If we remove only the original water in Pond , using pumps takes days; using pump takes days.
Thus, each pump removes of Pond ’s original water per day.
(2) Since Pond is times the size of Pond , removing its daily rainfall requires pumps.
Also, one pump removes of Pond ’s original water per day.
To remove it in days, we need: pumps.
Therefore, to remove both the rainfall and the original water from Pond in days requires: pumps.
Answer: .
