In a group of animals consisting of some buffaloes and some ducks, the total number of legs is 24 more than twice the total number of heads. 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 heads and legs problem that is frequently seen in school level mathematics and competitive exams. It involves forming linear equations using the number of animals, the number of heads, and the number of legs. The question combines logical reasoning with simple algebra to determine how many animals of each type are present. Here the focus is on the number of buffaloes in a mixed group of buffaloes and ducks.

Given Data / Assumptions:

• The group contains buffaloes and ducks only.
• Each animal has exactly 1 head.
• Each buffalo has 4 legs.
• Each duck has 2 legs.
• Total number of legs is 24 more than twice the total number of heads.
• All heads and legs are counted correctly.

Concept / Approach:
Let the number of buffaloes and ducks be variables. The total number of heads is simply the sum of these numbers, since each animal contributes one head. The total number of legs can be expressed as 4 times the number of buffaloes plus 2 times the number of ducks. The key relationship is that the total legs count exceeds twice the number of heads by exactly 24. This condition leads to a single linear equation in the number of buffaloes, which can then be solved directly.

Step-by-Step Solution:
Let the number of buffaloes be b. Let the number of ducks be d. Total number of heads = b + d. Total number of legs = 4b + 2d. According to the problem, total legs = 2 * (total heads) + 24. So, 4b + 2d = 2 * (b + d) + 24. Expand the right side: 4b + 2d = 2b + 2d + 24. Subtract 2b and 2d from both sides: 4b + 2d - 2b - 2d = 24. This simplifies to 2b = 24. So, b = 24 / 2 = 12. Therefore, there are 12 buffaloes in the group.

Verification / Alternative check:
We can verify the result by expressing d in terms of b. The equation 2b = 24 gives b = 12, but it does not restrict d directly. For any non negative number of ducks d, the relation 4b + 2d = 2(b + d) + 24 will hold only when b = 12. For example, if we assume there are 12 buffaloes and 6 ducks, then total heads = 18 and total legs = 12 * 4 + 6 * 2 = 48 + 12 = 60. Twice the number of heads is 36 and 36 + 24 = 60, so the condition is satisfied. The exact number of ducks is not required to answer the question, because the equation fixes only the number of buffaloes.

Why Other Options Are Wrong:
6 buffaloes: If b = 6, then 2b = 12 and 4b + 2d cannot be equal to 2(b + d) + 24 for any integer d that makes sense, so the relation fails.
8 buffaloes or 10 buffaloes: Substituting these values into the equation 4b + 2d = 2(b + d) + 24 shows that the left and right sides cannot be made equal for non negative integer d in a realistic animal group.
14 buffaloes: If b = 14, then 2b = 28, which forces a mismatch in the legs condition. Only 12 buffaloes satisfy the given relationship.

Common Pitfalls:
One common mistake is to forget that each animal contributes exactly one head, which makes the total heads easy to express. Some learners also confuse the statement and try to equate total legs directly to 24 or to twice the heads without the plus 24 term. Another error is manipulating the algebra incorrectly when simplifying 4b + 2d = 2b + 2d + 24. Careful step by step algebra helps avoid losing terms or cancelling incorrectly.

Final Answer:
The number of buffaloes in the group is 12 buffaloes.