To Experts - Any shortcut to this problem

As soon as you see that 42 works by multiplying the 10 fish, and everything else, by 2, you know that 84, or 2 x 42, works by multiplying the 5 dogs by 4.

As soon as you see that 105 works by multiplying the 4 cats, and everything else, by 5, you know that 210, or 2 x 105, works by multiplying the 2 birds, and everything else, by 10.

Alternatively, if you notice that you have 4x + 5x + 2x + 10x = 21x animals, and that 63 = 3 * 21, you might immediately see that 3 * any of the number of any of the types of animals will not be 20. So the one that won't work has to be 63.

Mo2men wrote:The ratio of cats to dogs to birds to fish is 4:5:2:10. There are no other types of animals in the pet store. If there are 20 of one type of animal at the store then which of the following cannot be the total number of animals in the store?

A) 42
B) 63
C) 84
D) 105
E) 210

C : D : B : F = 4:5:2:10.
Since the sum of the values in the ratio = 4+5+2+10 = 21, the total number of animals must be a multiple of 21.
The answer choices imply the following multipliers for the ratio:

A: 42 = 21*2
Here, the multiplier of 2 doubles the values in the ratio to C : D : B : F = 8:10:4:20, with the result that F=20.

B: 63 = 21*3.
Here, the multiplier of 3 triples the values in the ratio to C : D : B : F = 12:15:6:30, with the result that NONE of the values = 20.
Thus, the total number of animals CANNOT be 63.

The correct answer is B.

Mo2men wrote:The ratio of cats to dogs to birds to fish is 4:5:2:10. There are no other types of animals in the pet store. If there are 20 of one type of animal at the store then which of the following cannot be the total number of animals in the store?

A) 42
B) 63
C) 84
D) 105
E) 210

We are given that the ratio of cats to dogs to birds to fish is 4 : 5 : 2 : 10 = 4x : 5x : 2x : 10x.

Thus, the total number of animals is 21x.

Let’s analyze each answer choice:

A) 42

21x = 42

x = 2

Since 20 is a multiple of 2 and, in particular, the number of fish = 10x = 10(2) = 20, there can be 42 animals in the store.

B) 63

21x = 63

x = 3

Since 20 is not a multiple of 3, there cannot be 63 animals in the store.

Alternate solution:

We are given that the ratio of cats to dogs to birds to fish is 4 : 5 : 2 : 10 = 4x : 5x : 2x : 10x.

Thus, the total number of animals is 21x.

Since the number of one type of animal is 20, we can see that x could be 5, 4, 10, and 2, respectively, for the number of cats, dogs, birds, and fish to be 20.

Since the total number of animals is 21x, the total number of animals could be 105, 84, 210, and 42. We see that all of these 4 numbers are in the given answer choices. However, the answer choice of 63 is not one of these 4 numbers, so it can’t be the total number of animals.

Answer: B