A cage contains only peacocks and deer. Altogether there are 72 heads and 200 feet inside the cage. How many peacocks and how many deer are there?

Introduction / Context:
This is a classic heads and legs problem, a type of system of equations question often seen in aptitude tests. The cage contains only two types of animals, peacocks and deer, with different numbers of legs. We are given the total number of heads and feet and asked to find how many animals of each type are present. Such problems reinforce translating word statements into linear equations and solving them.

Given Data / Assumptions:

• The cage contains only peacocks and deer, no other animals.
• Each animal has exactly one head.
• A peacock has 2 feet.
• A deer has 4 feet.
• Total number of heads = 72.
• Total number of feet = 200.

Concept / Approach:
Let p be the number of peacocks and d be the number of deer. Since each animal contributes one head, the head count equation is p + d = 72. For feet, each peacock contributes 2 feet and each deer contributes 4 feet, so the feet equation is 2p + 4d = 200. We solve this system of linear equations to find p and d. Using substitution or elimination methods works well here.

Step-by-Step Solution:
Step 1: Let p = number of peacocks and d = number of deer.Step 2: From the number of heads, we get equation (1): p + d = 72.Step 3: From the number of feet, we get equation (2): 2p + 4d = 200.Step 4: Simplify equation (2) by dividing by 2: p + 2d = 100. Call this equation (3).Step 5: Subtract equation (1) from equation (3): (p + 2d) - (p + d) = 100 - 72.Step 6: This simplifies to d = 28.Step 7: Substitute d = 28 back into equation (1): p + 28 = 72, which gives p = 72 - 28 = 44.Step 8: Therefore there are 44 peacocks and 28 deer.

Verification / Alternative check:
Check the solution directly with the original data. Total heads: 44 + 28 = 72, which matches the given head count. Total feet: peacocks contribute 44 * 2 = 88 feet, and deer contribute 28 * 4 = 112 feet. The total is 88 + 112 = 200 feet, matching the given total. Since both conditions are satisfied, the solution is correct.

Why Other Options Are Wrong:
Option 42 peacocks and 24 deer: Heads = 66, so this does not match the required 72 heads.Option 38 peacocks and 26 deer: Heads = 64, which is too few.Option 36 peacocks and 24 deer: Heads = 60, also incorrect.Option 40 peacocks and 32 deer: Heads = 72, but feet = 40 * 2 + 32 * 4 = 80 + 128 = 208, not 200.

Common Pitfalls:
Students sometimes mix up the coefficients in the leg equation, assigning 4 legs to peacocks and 2 to deer, which leads to wrong answers. Others forget to divide the leg equation by a common factor to simplify it, making the arithmetic more error prone. A clear definition of variables and careful translation of the word problem into equations helps avoid these issues.

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