SOLUTION: a mathematician turned farmer has cows and chickens. he tells you that among the cows and chickens, there are 148 legs and 60 heads. how many cows and chickens on his farm?

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Question 1131674: a mathematician turned farmer has cows and chickens. he tells you that among the cows and chickens, there are 148 legs and 60 heads. how many cows and chickens on his farm?
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39615) About Me  (Show Source):
You can put this solution on YOUR website!
b cows (bovine)
p chickens (poultry)

system%284b%2B2p=148%2Cb%2Bp=60%29

system%282b%2Bp=74%2Cb%2Bp=60%29

E1-E2----------- highlight%28b=14%29------fourteen cows

highlight%28p=46%29-----fourtysix chickens

Answer by ikleyn(52768) About Me  (Show Source):
You can put this solution on YOUR website!
.
A mathematician turned farmer has cows and chickens.
He tells you that among the cows and chickens, there are 148 legs and 60 heads. How many cows and chickens highlight%28are%29 highlight%28there%29 on his farm?
~~~~~~~~~~~~~~~~~~~

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 chicken at the farm.
If you count the heads you get the equation
x + y = 60.
If you count the legs you get the equation
4x + 2y = 148.

So you have the system of two equations with two unknowns
system+%28x+%2B+y+=+60%2C%0D%0A4x+%2B+2y+=+148%29%0D%0A

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,
system+%282x+%2B+2y+=+120%2C%0D%0A4x+%2B+2y+=+148%29%0D%0A

4x - 2x = 148 - 120,
2x = 28,
x = 14.

So,  there are  14  cows in the farm.
Hence,  the number of chicken is  60 - 14 = 46.

Let us check the total number of legs.  You have altogether
4*14 + 2*46 = 148 legs.       ! Correct !

Answer.  There are  14  cows and  46  chicken at the farm.


Solution 2

Let  x  be the number of cows at a farm.
Then the number of chicken is  60 - x  in accordance with the condition.
If you count the legs you get the equation
4x + 2*(60-x) = 148.

To solve this equation open the parentheses and combine like terms,  step by step:
4x + 2*60 - 2x = 148,
2x + 120 = 148,
2x = 148 - 120,
2x = 28,
x = 14.

So,  there are  14  cows at the farm.
Hence,  the number of chicken is  60 - 14 = 46.

You got 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  60*2 = 120 legs.

This number is in  28  less than  148  legs given by condition.
Certainly,  these  24  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  28/2 = 14.
Hence,  the number of chicken is  60 - 14 = 46.

You get the same answer as in the solutions  1  and  2  above.

Answer.  There are  14  cows and  46  chicken at the farm.


For many other similar solved problems see the lessons
    - Problem on two-wheel and three-wheel bicycles
    - Problem on animals at a farm
    - Problem on pills in containers
    - What type of problems are these?
in this site.