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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