Question

A man has a certain number of chickens and goats. Their head count is \[30\]. If the total number of their legs is \[84\], what is the ratio between the number of chicken and goats?

Answer 1

Hint: Using the information about head and legs we will find the equation of numbers of chickens and goats by assigning different variables for number of chickens and number of goats. Solving these equations we will get the numbers of chickens and goats.

Complete step-by-step answer:
It is given that a man has a certain number of chickens and goats. The headcount is \[30\]. Total number of their legs is \[84\].
We have to find the ratio of the number of chicken and goats.
Before that we will find the number of chickens and goats.
Let us consider, the man has \[x\] number of chicken and \[y\] number of goats.
According to the problem,
\[x + y = 30\] … (1)
We know that a chicken has two legs and a goat has four legs.
So, \[x\] number of chickens have \[2x\] legs and \[y\] number of goats have \[4y\].
So, according to the next condition given in the problem we have,
\[2x + 4y = 84\]
Let us divide \[2\] on both the sides of the above equation then we have,
\[x + 2y = 42\]… (2)
Let us subtract equation (2) by (1) we get,
\[x + 2y - x - y = 42 - 30\]
Solving we have,
\[y = 12\]
So, the number of goats in the farm is \[12\].
The number of chicken is found from equation (1)
which implies the number of chicken is \[30 - 12 = 18\].
Hence, the man has \[12\] goats and \[18\] chickens.
Hence the ratio between the number of chicken and goats is \[18:12 = 3:2\]
Hence the man has chicken and goats in the ratio \[3:2\]

Note: To solve two variables we need two equations. There are many other methods to solve these equations but for every method the solution will always be the same always.