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This problem trains for: SAT-I, AMC-8, GMAT.
Two kittens had lapped up half of the food in the bowl at the time a third kitten showed up and joined in the eating. If all the kittens eat at the same rate and it takes a total of 10 s to empty the bowl, how many seconds after the start did the third kitten show up?
Denote the unknown time with x. Then, eating the first half of the food took x seconds and eating the second half took 10-x seconds.
If each kitten eats r units of food per second, two kittens eat 2r and three kittens eat 3r units of food per second.
Since the amount eaten in the first period is equal to the amount eaten in the second period (half the food), then we can write:
Therefore, the first half gets eaten in 6 seconds and this is also how long after the start the third kitten joined in. It has taken all three only 4 seconds to finish off the food.