Lesson Problem on animals at a farm

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<H2>Problem on animals at a farm</H2>

<H3>Problem</H3>There are rabbits and turkeys at a farm, &nbsp;41&nbsp; heads and &nbsp;132&nbsp; legs in all.
There is no need to explain that a rabbit has four legs, &nbsp;while a turkey has two legs. 
How many rabbits and turkeys are at the farm?

You can solve this problem in three different ways. 
If you are familiar with systems of linear equations, &nbsp;you can reduce the problem to the system of two linear equations in two unknowns and solve it.
This way is implemented in the &nbsp;Solution 1&nbsp; below.
You can also reduce the problem to one equation with one unknown and solve it. &nbsp;This is done in the &nbsp;Solution 2&nbsp; below.
Alternatively, &nbsp;you can solve the problem simply applying logical reasoning and not using equations at all. &nbsp;This is done in the &nbsp;Solution 3&nbsp; below.

Solution 1

Let &nbsp;x&nbsp; be the number of rabbits and &nbsp;y&nbsp; be the number of turkeys at the farm.
If you count the heads you get the equation
x + y = 41.
If you count the legs you get the equation 
4x + 2y = 132.

So you have the system of two equations with two unknowns
{{{system (x + y = 41,
4x + 2y = 132)
}}}

To solve this system of equations multiply the first equation by &nbsp;2&nbsp; and subtract the obtained equation from the second one. 
You will get, step by step,
{{{system (2x + 2y = 82,
4x + 2y = 132)
}}}

4x - 2x = 132 - 82,
2x = 50,
x = 25.

So, &nbsp;there are &nbsp;25&nbsp; rabbits in the farm. 
Hence, &nbsp;the number of turkeys is &nbsp;41 - 25 = 16.

Let us check the total number of legs. &nbsp;You have altogether
4*25 + 2*16 = 100 + 32 = 132 legs.

Answer. &nbsp;There are &nbsp;25&nbsp; rabbits and &nbsp;16&nbsp; turkeys at the farm.

Solution 2

Let &nbsp;x&nbsp; be the number of rabbits at a farm.
Then the number of turkeys is &nbsp;41 - x&nbsp; in accordance with the condition.
If you count the legs you get the equation 
4x + 2*(41-x) = 132.

To solve this equation open the brackets and combine like terms, &nbsp;step by step:
4x + 2*41 - 2x = 132,
2x + 82 = 132,
2x = 132 - 82,
2x = 50,
x = 25.

So, &nbsp;there are &nbsp;25&nbsp; rabbits at the farm. 
Hence, &nbsp;the number of turkeys is &nbsp;41 - 25 = 16.

Let us check the total number of legs. &nbsp;You have altogether
4*25 + 2*16 = 100 + 32 = 132 legs.

You get the same answer as in the Solution 1.

Answer. &nbsp;There are &nbsp;25&nbsp; rabbits and &nbsp;16&nbsp; turkeys 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 &nbsp;41*2 = 82 legs.

This number is in &nbsp;50&nbsp; less than &nbsp;132 &nbsp;legs given by condition. 
Certainly, &nbsp;these &nbsp;50&nbsp; legs belong to rabbits in the number of &nbsp;2&nbsp; legs to each rabbit &nbsp;(in addition to that two legs we just counted under the assumption). 
This means that the number of rabbits is &nbsp;50/2 = 25.
Hence, &nbsp;the number of turkeys is &nbsp;41 - 25 = 16.

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

Answer. &nbsp;There are &nbsp;25&nbsp; rabbits and 16 turkeys at the farm.

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