There are 100 dogs at pampered paws kennel. each day, the dogs eat 100 lb of food. large dogs eat 3 lb, medium dogs eat 2 lb, and small dogs eat 0.5 lb. how many of each dog are at the kennel if there are five times as many medium dogs as large ones?

The answer is 5 large, 25 medium, 70 small dogs.

x - the number of large dogs
y - the number of medium dogs
z - the number of small dogs

There are 100 dogs at Pampered Paws Kennel:
x + y + z = 100

Each day, the dogs eat 100 lb of food. Large dogs eat 3 lb, medium dogs eat 2 lb, and small dogs eat 0.5 lb:
3x + 2y + 0.5z = 100

There are five times as many medium dogs as large ones:
x = 5y

x + y + z = 100
3x + 2y + 0.5z = 100
y = 5x

Substitute y in the first two equations:
x + 5y + z = 100
3x + 2*5x + 0.5z = 100
___
6x + z = 100
13x + 0.5z = 100

z = 100 - 6x
13x + 0.5z = 100

13x + 0.5(100 - 6x) = 100
13x + 50 - 3x = 100
13x - 3x = 100 - 50
10x = 50
x = 50/10 = 5

y = 5x = 5 * 5 = 25

x + y + z = 100
5 + 25 + z = 100
30 + z = 100
z = 100 - 30
z = 70

Logical equivalent of the give question is its contrapositive