*a Student sees rabbits, ducks and birds in the park. she counts 22 heads and 52 feet. students know that the number of birds is 2 more than the sum of the rabbits and ducks. how many of each animal is there. ( all animals are normal )?*

a Student sees rabbits, ducks and birds in the park. she counts 22 heads and 52 feet. students know that the number of birds is 2 more than the sum of the rabbits and ducks. how many of each animal is there. ( all animals are normal )?

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Hi Nancy;

22=R+B+D

B=R+D+2

52=4R+2B+2D

The last equation can also be...

26=2R+B+D

Let's consider the following equations...

26=2R+B+D

22= R+B+D

Let's subtract the second equation from the first...

4=R

22=R+B+D

Let's substitute B with R+D+2

22=R+(R+D+2)+D

22=2R+2D+2

Let's substitute R with 4...

22=[2(4)]+2D+2

22=8+2D+2

Let's combine like terms...

22=2D+10

Let's subtract 10 from both sides...

22-10=D+10-10

12=2D

Let's divide both sides by 2...

12/2=2D/2

6=D

22=R+B+D

22=4+B+6

22=B+10

12=B

4 Rabbits

6 Ducks

12 Birds

Let's check our work...

Does

4+6+12=22?

22=22--YES!

Does

4(4)+2(6)+2(12)=52?

16+12+24=52

52=52--YES!

Does

12=4+6+2

12=12--YES!

Nancy, the key is to take each idea and recognize that it creates an equation of some kind, and that the story needs to create as many equations as variables.

First, your problem involves three animals that we need to determine the numbers of. So let's assign variables

r=# of rabbits

d=# of ducks

b=# of birds (I'm going to assume that the birds category excludes ducks).

Second, let's assume each bird and duck has 1 head and 2 feet and a rabbit has 1 head and 4 feet.

Third, we are told that there are 22 heads. Since each animal has 1 head, we know that

(1 x r) + (1 x d) + (1 x b) = 22

Fourth, we are told that there are 52 feet, since ducks and birds have 2 and rabbits have 4 feet, we know that

(4 x r) + (2 x d) + (2 x b) = 52

Fifth, we need to translate "the number of birds is 2 more than the sum of the rabbits and ducks"

The sum of rabbits and ducks translates to (r + d). Two more that that sum is ((r + d) + 2) and that has to equal the number of birds, so b = (r + d) + 2

We now have three equations:

r + d + b = 22

4r + 2d + 2b = 52

b = r + d + 2

If you need help using substitution to solve 3-equations and 3-unknowns like these, please do not hesitate to ask for more help; otherwise, I'll leave it to you.

Appreciate all your good explanations as to setting up the problem so one could find an answer. Thanks!

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