In a group of cows and hens, the number of legs is 14 more than twice the…

Concept

Each animal contributes exactly 1 head, so the head count equals the total number of animals. A hen has 2 legs and a cow has 4 legs. Twice the number of heads counts 2 legs for every animal, so any amount by which the total legs exceed twice the heads must come solely from the 2 extra legs that each cow carries beyond a hen.

Application

1. Let the number of cows be C and the number of hens be H. Then heads = C + H, and legs = 4C + 2H.

2. The condition states legs = 2 × (heads) + 14, that is 4C + 2H = 2(C + H) + 14.

3. Expand the right side: 4C + 2H = 2C + 2H + 14.

4. Cancel 2H from both sides: 4C = 2C + 14, so 2C = 14.

5. Divide by 2 to get C = 7. The hen count cancels, so the answer does not depend on how many hens are present.

Cross-check

Take C = 7 and any number of hens, say H = 3. Then heads = 10 and legs = 4(7) + 2(3) = 34. Twice the heads is 20, and 34 − 20 = 14, exactly the required surplus. So 7 cows satisfies the condition for every choice of hens, confirming the result.