Question

A nonzero number of goats, each with four legs, and a nonzero number of chickens, each with two legs, are living on a farm. Between all the animals, there are as many heads as there are legs. Assuming that all animals have all of their limbs, what is the fewest possible number of chickens on the farm? A) 8 B) 16 C) 24 D) 32

Answer 1

The fewest possible number of chickens on the farm can be found by solving a system of equations. The number of chickens is 24. (Option C)

Step-by-step explanation:

The fewest possible number of chickens on the farm can be found by solving a system of equations.

Let's represent the number of goats as 'g' and the number of chickens as 'c'.

Each goat has four legs, so the total number of goat legs is 4g.

Each chicken has two legs, so the total number of chicken legs is 2c.

In total, there are as many heads as there are legs. Since goats and chickens each have one head, the total number of heads is g + c.

From the given information, we can set up the following equations:

4g + 2c = g + c

3g = -c

Since both the number of goats and chickens must be nonzero, the fewest possible number of chickens is multiple of 3 which is 24 based on options.

Answer 2

The fewest possible number of chickens on the farm is 24 (Option C).

Let's denote the number of goats as g and the number of chickens as c.

Each goat has 4 legs and 1 head, while each chicken has 2 legs and 1 head.

The total number of legs on the farm is 4g + 2c (since goats have 4 legs each and chickens have 2 legs each).

The total number of heads on the farm is g + c (since each goat and each chicken has 1 head).

According to the given information, the number of heads is equal to the number of legs:

g + c = 4g + 2c

Rearranging the equation, we get:

c = 3g

This equation tells us that the number of chickens (c) is three times the number of goats (g).

As both the number of chickens and goats need to be non-zero, let's consider the smallest positive integer values for g and c that satisfy c = 3g and are non-zero. The smallest positive integer values that satisfy this relationship are g = 8 and c = 24.