word problem

Question

At a pet hotel there are nightly loding rates. For small dogs $21, $26 for medium dogs, $31 for large dogs, and $14 for cats.  One night there were 31 guest. 

The guest included twice as many small dogs as medium dogs

Large dogs were the smallest number of animals in the hotel.

There were seven times as many cats as there were large dogs.

The hotel collected $598 in guest fees for the evening.

How many ofe ach kind and size of pet stayed at the hotel that night?

Answer 1

There are two basic equations to consider in this problem: Number of guests and income received from the guests. Number of Guests: Small Dogs + Medium Dogs + Large Dogs + Cats = 31 Income: (Small Dogs)(21) + (Medium Dogs)(26) + (Large Dogs)(31) + (Cats)(14) = 598 The problem tells us that there were twice as many small dogs as medium, so let's let X = # of medium dogs, then 2X = # of small dogs. We also know there were 7 times as many cats as large dogs, so Y = # of large dogs 7Y = # of cats. Putting these into our equations, we get 2X + X + Y + 7Y = 31 (2X)(21) + (X)(26) + (Y)(31) + (7Y)(14) = 598 Simplifying both we get 3X + 8Y = 31 42X + 26X + 31Y + 98Y = 68X + 129Y = 598 We now have 2 equations in 2 variables. Let's solve the first for Y and substitute that into the second equation: 8Y = 31 - 3X Y = (31 - 3X)/8 68X + 129(31 - 3x)/8 = 598 Multiply both sides by 8 to clear the denominator: (8)68X + 129(31 - 3X) = (8)(598) 544X + 3999 - 387X = 4784 Simplifying, 157X + 3999 = 4784 157X = 785 X = 5 From this, we know there were 5 Medium dogs and 10 small dogs. We plug this back into our first equation and solve for Y: 2(5) + (5) + Y + 7Y = 31 10 + 5 + 8Y = 31 15 + 8Y = 31 8Y = 16 Y = 2 So there were 2 large dogs, and 7Y = (7)(2) = 14 cats. So the answer is: 10 Small Dogs 5 Medium Dogs 2 Large Dogs 14 Cats.

Answer 2

You can try guess-and-check approach. The problem says that "Large dogs were the smallest number of animals". Pick a small number (3 for example) for a number of large dogs. Use other pieses of information to find number of cats, medium dogs and small dogs. Then find the lodgin cost for all the animals. If found price is more then $598, reduce the number of large dogs; if it less then $598, increace the number of large dogs. Try again untill lodging price will be exacly $598. Good luck.