Shortcut Trick
If all 60 animals were penguins (2 feet), the total number of feet would be 60 × 2 = 120.
The extra feet given in the question = 160 − 120 = 40.
Each polar bear contributes an extra 2 feet (since 4 − 2 = 2).
Number of polar bears = Total extra feet ÷ Extra feet per bear = 40 ÷ 2 = 20.
∴ The correct answer is 20 polar bears.
Alternate Method
Given:
⇒ Total number of heads = 60
⇒ Total number of feet = 160
⇒ Feet per penguin = 2
⇒ Feet per polar bear = 4
Method 1: Alligation Approach
⇒ If all animals were penguins, total feet = 60 × 2 = 120.
⇒ If all animals were polar bears, total feet = 60 × 4 = 240.
⇒ The average total feet given is 160.
Penguins (2 feet)
Polar Bears (4 feet)
120
240
160
80
40
Ratio → 80 : 40 = 2 : 1
⇒ The ratio of penguins to polar bears is 2 : 1.
⇒ Total ratio parts = 2 + 1 = 3.
⇒ Number of polar bears = (1 ÷ 3) × 60 = 20.
Method 2: Linear Equation
⇒ Let the number of penguins be P and polar bears be B.
⇒ Total heads: P + B = 60 ... (Equation 1)
⇒ Total feet: 2P + 4B = 160
⇒ Dividing by 2: P + 2B = 80 ... (Equation 2)
⇒ Subtracting (1) from (2): (P + 2B) − (P + B) = 80 − 60.
⇒ B = 20.
∴ The correct answer is 20 polar bears.
Additional Information
Heads and Legs Direct Formula
For mixtures of 2-legged and 4-legged animals, the exact number of 4-legged animals can be directly found using: (Total Legs ÷ 2) − Total Heads.
Alligation Cross Method
It is a quick ratio determination technique used when two discrete properties are mixed to produce a known overall average value. Subtract diagonally to find the base ratio components.
Linear Algebraic Elimination
Forming two basic simultaneous equations is a universal rule for standard replacement and count problems to ensure maximum accuracy if shortcuts are forgotten.