In a zoo there are rabbits and pigeons. The total number of heads is 340 and the total number of legs is 1060. How many pigeons are there?

Introduction / Context:
This is a classic heads and legs puzzle used to test linear equation modelling. Rabbits and pigeons have different numbers of legs, and the problem asks you to deduce how many pigeons are present using total head and leg counts.

Given Data / Assumptions:

• There are only two types of animals: rabbits and pigeons.
• Total number of heads = 340.
• Total number of legs = 1060.
• Each rabbit has 4 legs.
• Each pigeon has 2 legs.
• We must find the number of pigeons.

Concept / Approach:
Let the number of rabbits and pigeons be variables. Each animal contributes one head, but different numbers of legs. We can write one equation for heads and one for legs, and then solve the resulting system of two linear equations. This is a direct application of algebra to a word problem.

Step-by-Step Solution:
Step 1: Let r be the number of rabbits and p be the number of pigeons. Step 2: From heads, we have r + p = 340. Step 3: From legs, rabbits contribute 4r legs and pigeons contribute 2p legs, so 4r + 2p = 1060. Step 4: Divide the leg equation by 2 to simplify: 2r + p = 530. Step 5: Now subtract the head equation (r + p = 340) from 2r + p = 530. Step 6: This gives (2r + p) - (r + p) = 530 - 340, so r = 190. Step 7: Substitute r = 190 into r + p = 340 to get 190 + p = 340, hence p = 150.

Verification / Alternative check:
Check the leg count with r = 190 and p = 150. Total legs = 190 * 4 + 150 * 2 = 760 + 300 = 1060, which matches the given number of legs. The head count is 190 + 150 = 340, also matching the given heads. Both conditions are satisfied, confirming that p = 150 is correct.

Why Other Options Are Wrong:
If pigeons were 120, 170, 180 or 190, the corresponding rabbit count from r + p = 340 would lead to a different total leg count that does not equal 1060. For example, with 170 pigeons, rabbits would be 170 and legs would sum to 170 * 4 + 170 * 2 = 1020, which is incorrect.

Common Pitfalls:
Some learners try to guess combinations without writing equations, which can be time consuming and error prone. Others mistakenly assign 2 legs to rabbits and 4 to pigeons. Another frequent error is forgetting to divide the leg equation by 2 to simplify, which makes the algebra unnecessarily complicated.

Final Answer:
The number of pigeons in the zoo is 150.