In a group consisting only of buffaloes and ducks, the total number of legs is 24 more than twice the total number of heads. Assuming each buffalo has 4 legs and each duck has 2 legs, how many buffaloes are there in the group?

Introduction / Context:
This is a classic word problem involving animals, heads and legs, which translates into a pair of simultaneous equations. Such problems test your ability to convert a real-world description into algebraic expressions and then solve for the required quantities, in this case the number of buffaloes.

Given Data / Assumptions:

• The group consists only of buffaloes and ducks.
• Each buffalo has 1 head and 4 legs.
• Each duck has 1 head and 2 legs.
• The total number of legs is 24 more than twice the total number of heads.

Concept / Approach:
Let the number of buffaloes be b and the number of ducks be d. The total number of heads is b + d. The total number of legs is 4b + 2d. The condition "legs are 24 more than twice the heads" becomes an equation linking these quantities. We then solve for b, which is the required answer.

Step-by-Step Solution:
Total heads = b + d.Total legs = 4b + 2d.Twice the number of heads is 2(b + d) = 2b + 2d.The condition says: total legs = twice heads + 24. So 4b + 2d = 2b + 2d + 24.Subtract 2b + 2d from both sides: 2b = 24, so b = 12.

Verification / Alternative check:
If b = 12, let us check the relationship for any value of d. Heads = 12 + d, twice heads = 2(12 + d) = 24 + 2d.Legs = 4b + 2d = 4(12) + 2d = 48 + 2d.Difference between legs and twice heads is (48 + 2d) − (24 + 2d) = 24, which matches the given condition regardless of how many ducks are present.

Why Other Options Are Wrong:
If b = 6, then the leg equation would become 4(6) + 2d = 2(6 + d) + 24, leading to 24 + 2d = 12 + 2d + 24, which simplifies to 24 = 36, a contradiction.Similarly, choosing 18 or 24 for b would not satisfy the derived algebraic condition 2b = 24, so they cannot be correct.

Common Pitfalls:
A common error is to misinterpret "24 more than twice the number of heads" and write 4b + 2d + 24 = 2(b + d) instead of 4b + 2d = 2(b + d) + 24.Another pitfall is to try random combinations of buffaloes and ducks without forming equations, which is inefficient and prone to mistakes.

Final Answer:
The group must contain 12 buffaloes to satisfy the given condition.