Question 1092717: If you have 23 heads and 72 total legs, How many cows and roosters do you have? I have already tried 12 cows to get 48 legs and then had 11 roosters to add 22 legs. (A cow has 4 legs and a rooster has 2)
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39623)   (Show Source):
Answer by ikleyn(52835)  (Show Source):
You can put this solution on YOUR website! If you have 23 heads and 72 total legs, How many cows and roosters do you have?
(A cow has 4 legs and a rooster has 2)
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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 cows and y be the number of roosters at the farm.
If you count the heads you get the equation
x + y = 23.
If you count the legs you get the equation
4x + 2y = 72.
So you have the system of two equations with two unknowns
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,
4x - 2x = 72 - 46,
2x = 26,
x = 13.
So, there are 13 cows in the farm.
Hence, the number of roosters is 23 - 13 = 10.
Let us check the total number of legs. You have altogether
4*13 + 2*10 = 52 + 20 = 72 legs.
Answer. There are 13 cows and 10 roosters at the farm.
Solution 2
Let x be the number of cows at a farm.
Then the number of roosters is 23 - x in accordance with the condition.
If you count the legs you get the equation
4x + 2*(23-x) = 72.
To solve this equation open the brackets and combine like terms, step by step:
4x + 2*23 - 2x = 72,
2x + 46 = 72,
2x = 72 - 46,
2x = 26,
x = 13.
So, there are 13 cows at the farm.
Hence, the number of roosters is 23 - 13 = 10.
Let us check the total number of legs. You have altogether
4*13 + 2*10 = 52 + 20 = 72 legs.
You get the same answer as in the Solution 1.
Answer. There are 13 cows and 10 roosters at the farm.
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 23*2 = 46 legs.
This number is 26 = 72 - 46 less than 72 legs given by condition.
Certainly, these 26 legs belong to cows in the number of 2 legs to each cow (in addition to that two legs we just counted under the assumption).
This means that the number of cows is 26/2 = 13.
Hence, the number of roosters is 23 - 13 = 10.
You get the same answer as in the solutions 1 and 2 above.
Answer. There are 13 cows and 10 roosters at the farm.
This kind of problems are traditionally considered as entertainment problems.
And they traditionally are used to show the students all three approaches.
To see more similar problems of this kind and their solutions, look into the lessons
- Problem on animals at a farm (*)
- Problem on two-wheel and three-wheel bicycles
- Problem on pills in containers and
- What type of problems are these?
in this site.
Notice that the lesson (*) has practically identical problem.
It is because of tradition again.
For this post, I simply made the global replacement "rabbits ---> cows" and "turkey ---> roosters" in that text file
(and changed the numbers respectively).
Ha-ha-ha.
H a p p y l e a r n i n g ! !
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