In a cage there are only peacocks and deer. Altogether there are 72 heads and 200 feet. Assuming each bird or animal is healthy and normal, how many peacocks and how many deer are there in the cage?

Introduction:
This is a classic heads and feet puzzle, often used to test linear equations and logical reasoning. You must use the fact that all animals have one head, but peacocks and deer have different numbers of legs, to find the exact count of each type of animal in the cage.

Given Data / Assumptions:

• Total number of heads = 72.
• Total number of feet = 200.
• Each peacock has 1 head and 2 legs.
• Each deer has 1 head and 4 legs.
• There are only peacocks and deer in the cage.

Concept / Approach:
We assign variables to represent the number of peacocks and deer, then form two linear equations. One equation comes from counting heads; the other comes from counting feet. Solving this system of equations gives the number of each type of animal.

Step-by-Step Solution:
Step 1: Let P be the number of peacocks and D be the number of deer. Step 2: From the heads count, each animal has one head: P + D = 72. Step 3: From the feet count, peacocks have 2 legs and deer have 4 legs: 2P + 4D = 200. Step 4: Simplify the feet equation by dividing by 2: P + 2D = 100. Step 5: Subtract the heads equation from this: (P + 2D) − (P + D) = 100 − 72, which gives D = 28. Step 6: Substitute D = 28 into P + D = 72: P + 28 = 72, so P = 72 − 28 = 44.

Verification / Alternative check:
Check with the feet count: 44 peacocks contribute 44 * 2 = 88 feet, and 28 deer contribute 28 * 4 = 112 feet. Total feet = 88 + 112 = 200, which matches the given value. Thus the solution is consistent.

Why Other Options Are Wrong:

• 42 and 24: Heads = 66, feet = 2*42 + 4*24 = 84 + 96 = 180, not correct.
• 38 and 26: Heads = 64, feet = 76 + 104 = 180, not 200.
• 36 and 24: Heads = 60, feet = 72 + 96 = 168, not 200.
• 40 and 32: Heads = 72, but feet = 80 + 128 = 208, which is too many.

Common Pitfalls:
Learners sometimes mix up the leg counts for each animal or forget to divide the feet equation to simplify. It is also easy to miscalculate totals if you do not check both equations. Always verify your solution against both the heads and the legs conditions.

Final Answer:
There are 44 peacocks and 28 deer in the cage.