Farm count — in a mixed group of cows and hens, the total legs are 14 more than twice the total heads. How many cows are there?

Introduction / Context:
This classic heads-and-legs puzzle uses simple linear equations. Cows have 4 legs, hens have 2 legs. Heads count equals the total number of animals. A relation connects legs to twice the number of heads with a fixed surplus of 14 legs.

Given Data / Assumptions:

• Cows = c; Hens = h.
• Heads = c + h.
• Legs = 4c + 2h.
• Condition: 4c + 2h = 2(c + h) + 14.

Concept / Approach:
Set up the equation using the leg and head counts. Simplify to isolate the number of cows. Only one variable will remain after cancellation because both sides include 2h.

Step-by-Step Solution:

Verification / Alternative check:
Pick any h and verify the relation holds by computing heads and legs. The equation above guarantees correctness for c = 7 regardless of h, consistent with the given relation structure.

Why Other Options Are Wrong:

• 5, 10, 12 — substituting these values does not satisfy 4c + 2h = 2(c + h) + 14 for all h; only c = 7 fits the simplified equation.

Common Pitfalls:
Forgetting that the number of heads equals total animals. Also, do not overcomplicate by trying to find h explicitly; the equation collapses neatly to c = 7.

Final Answer:
7