.
During a certain ceremony goats and chicken were slaughtered.The number of heads(for both chicken and goats) was 45.
The total number of legs was 100. Determine the exact number of goats and chicken slaughtered.
~~~~~~~~~~~~~~~~~~~~~~
It is WELL KNOWN entertainment problem,
and it deserves an adequate treatment.
You can solve this problem in three different ways.
If you are familiar with systems of linear equations, you can reduce the problem to the system of two linear equations in two unknowns
and solve it. This way is implemented in the Solution 1 below.
You can also reduce the problem to one equation with one unknown and solve it. This is done in the Solution 2 below.
Alternatively, you can solve the problem simply applying logical reasoning and not using equations at all. This is done in the Solution 3 below.
Solution 1
Let x be the number of goats and y be the number of chicken at the farm.
Then you have a system of two equations
x + y = 45, (1) (counting heads)
4x + 2y = 100. (2) (counting legs)
To solve this system of equations, multiply the first equation by 2 and subtract the obtained equation from the second one.
You will get, step by step,
2x + 2y = 90, (1')
4x + 2y = 100, (2')
4x - 2x = 100 - 90 ====> 2x = 10 ====> x = = 5.
So, there are 5 goats at the farm.
Hence, the number of chickens is 45 - 5 = 40.
Let us check the total number of legs. You have altogether
4*5 + 2*40 = 20 + 80 = 100 legs. ! Correct !
Answer. There are 5 goats and 40 chicken at the farm.
Solution 2
Let x be the number of goats at a farm.
Then the number of chicken is (45 - x), in accordance with the condition.
If you count the legs you get the equation
4x + 2*(45-x) = 100.
To solve this equation, open parentheses and combine like terms, step by step:
4x + 2*45 - 2x = 100,
2x + 90 = 100 ====> 2x = 100 - 90 ====> 2x = 10 ====> x = = 5.
So, there are 5 goats at the farm.
Hence, the number of chicken is 45 - 5 = 40.
Let us check the total number of legs. You have altogether
4*5 + 2*40 = 20 + 80 = 100 legs. ! Correct !
You get the same answer as in the Solution 1.
Solution 3
Let us suppose for a moment that all the animals at the farm have two legs each.
Under this assumption, the total number of legs is 45*2 = 90 legs.
This number is by 10 less than 100 legs given by condition.
Certainly, these 50 legs belong to goats in the number of 2 legs to each goat
(in addition to that two legs we just counted under the assumption).
This means that the number of goats is 10/2 = 5.
Hence, the number of chicken is 45 - 5 = 40.
You get the same answer as in the solutions 1 and 2 above.
For similar problems, see the lessons
Problem on two-wheel and three-wheel bicycles
Problem on pills in containers and
What type of problems are these?
under the current topic Miscellaneous Word Problems of the section Word problems in this site.
Use this file/link ALGEBRA-I - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-I.