lheiannie07 wrote:How many more dogs than cats are in the veterinarian's office?

(1) There is a total of 30 dogs and cats in the office.
(2) The number of cats is the square root of the number of dogs.

Let D = # of dogs
Let C = # of cats

Target question: What is the value of D - C?

Statement 1: There is a total of 30 dogs and cats in the office
In other words, D + C = 30
There are several scenarios that satisfy statement 1. Here are two:
Case a: D = 29 and C = 1. In this case, the answer to the target question is D - C = 29 - 1 = 28
Case b: D = 20 and C = 10. In this case, the answer to the target question is D - C = 20 - 10 = 10
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The number of cats is the square root of the number of dogs.
In other words, C = √D
There are several scenarios that satisfy statement 1. Here are two:
Case a: D = 9 and C = 3. In this case, the answer to the target question is D - C = 9 - 3 = 6
Case b: D = 25 and C = 5. In this case, the answer to the target question is D - C = 25 - 5 = 20
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
From both statements, we get the following system:
D + C = 30
C = √D

Take top equation and replace C with √D to get: D + √D = 30
Subtract D from both sides: √D = 30 - D
Square both sides: (√D)² = (30 - D)²
Expand: D = 900 - 60D + D²
Rearrange to get: D² - 61D + 900 = 0
Factor to get: (D - 25)(D - 36) = 0
So, EITHER D = 25 OR D = 36
From statement 1, we can see that there CANNOT be more than 30 dogs, so we can ignore the solution D = 36
So, it must be the case that D = 25
If D = 25, then C = 5
So, the answer to the target question is D - C = 25 - 5 = 20
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

lheiannie07 wrote:How many more dogs than cats are in the veterinarian's office?

(1) There is a total of 30 dogs and cats in the office.
(2) The number of cats is the square root of the number of dogs.

What's the best way to determine whether statement 1 is sufficient?

OA C

Say the number of dogs = x and the number of cats = y. We have to find out the value of x - y.

(1) There is a total of 30 dogs and cats in the office.

=> x + y = 30. We cannot find out the unique values of x and y, thus x - y. Insufficient.

(2) The number of cats is the square root of the number of dogs.

=> y = √x. We cannot find out the unique value of x and y, thus x - y. Insufficient.

(1) and (2) combined

Replacing the value of y = √x in x + y = 30, we get x + √x = 30.

We need not solve this complicated quadratic equation. Since the number of cats = y = √x is a positive integer, x must be a perfect square. Thus, x must be one among 1, 4, 9, 16, and 25. x cannot be 36 or greater since 36 > 30 (Total number of dogs and cats).

x: {1, 4, 9, 16, 25}

Let's do some hit and trial.

Case 1: Say x = 25, then √ x = √25 = 5. Thus x + √x = 30 => 25 + √25 ? 30 => 25 + 5 = 30. x can be 25.
Case 2: Say x = 16, then √ x = √16 = 4. Thus x + √x = 30 => 16 + √16 ? 30 => 16 + 4 = 20 ≠ 30. x cannot be 16. This is an invalid case.

There is no need to try x for 9, 4 and 1 and this will give even smaller values. Thus, x = 25 and y = 5 => x - y = 25 - 5 = 20.

The correct answer: C

Hope this helps!

-Jay
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