A Zookeeper Counted the Heads of the Animals in a Zoo and Found it to be 80 GMAT Problem Solving

Question: A zookeeper counted the heads of the animals in a zoo and found it to be 80. When he counted the legs of the animals he found it to be 260. If the zoo had either pigeons or horses, how many horses were there in the zoo?

1. 30
2. 40
3. 50
4. 60
5. 70

“A zookeeper counted the heads of the animals in a zoo and found it to be 80” – this topic is a part of GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical understanding.

Approach Solution 1: 

Given: A zookeeper counted the heads of the animals in a zoo and found it to be 80. 
When he counted the legs of the animals he found it to be 260.
We can derive that heads are 80 and legs are 260. Since, heads are 80, total number of animals are 80. 
Legs can differ as there are two legged and four legged animals. 
Also, the zoo has pigeons and horses only. 

Asked: If the zoo had either pigeons or horses, how many horses were there in the zoo?

Let the number of pigeons and horses in the zoo be p & h respectively

Since pigeons and horses each have only 1 head.
Pigeons + Horses = Total number of animals.
p + h = 80 --------- (1)

Since pigeons have 2 legs and horses have 4 legs each.
2*pegions + 4*horses = Total number of legs
2*p + 4*h = 260 --------- (2)
2 (p+2h) = 260
p+2h = 260/2
p + 2h = 130

Now, considering the above equations:

h = 130 - 80 = 50

There are 50 horses in the zoo. Hence, the answer C is correct. 

Correct Answer: C

Approach Solution 2:

This solution is based on the logic of "allegation and mixture".

Given: A zookeeper counted the heads of the animals in a zoo and found it to be 80. 
When he counted the legs of the animals he found it to be 260.
We can derive that heads are 80 and legs are 260. Since, heads are 80, total number of animals are 80. 
Legs can differ as there are two legged and four legged animals. 
Also, the zoo has pigeons and horses only. 

Asked: If the zoo had either pigeons or horses, how many horses were there in the zoo?
Heads Count = 80, and Legs Count = 260

Pigeons (+2 Legs)---------------------Horses (+4 Legs)
Let’s consider there are 80 each in the zoo.
2 x 80 = 160---------------------------4 x 80 = 320

Now, the problem statement says that there are 260 legs. So,
Considering both scenarios, 
320 - 260 = 60------------------------260 - 160 =100
60 : 100 is the ratio,
If we simplify this, the ratio comes down to:
3 : 5
Total is (3 + 5) 8 ---> 80
1 ---> 10
Pigeons = 10 x 3 = 30
Horses = 10 x 5 = 50

Hence, the correct answer is C.

Correct Answer: C

Approach Solution 3:

Let the number of horses be x
So, the number of pigeons is 80-x.
Then, the each pigeon has 2 legs and each horse has 4 legs.
Hence, the total number of legs =4x+2(80-×)=260

4x + 160 - 2x = 260

implies that 2x = 100

implies that x=50

Correct Answer: C

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