Your classmate is starting a new fitness program. He is planning to ride his bicycle 45 minutes every day. He burns 10 calories per minute bicycling at 11 mph and 13.75 calories per minute bicycling at 15 mph. How long should he bicycle at each speed to burn 600 calories?
Let x be the number of minutes he should ride at 11 mph.
Therefore, the number of minutes he should ride at 15 mph is 45 - x.
He will burn 10x calories riding at 11 mph.
He will burn 13.75(45 - x) calories riding at 15 mph.
The total calories he burns is 10x + 13.75(45 - x) = 600.
Expanding the equation, we get 10x + 618.75 - 13.75x = 600.
Combining like terms, we get -3.75x + 618.75 = 600.
Subtracting 618.75 from both sides, we get -3.75x = -18.75.
Dividing both sides by -3.75, we get x = 5.
He should ride at 11 mph for 5 minutes and at 15 mph for 45 - 5 = <<45-5=40>>40 minutes. Answer: \boxed{5, 40}.