If a farmer can trade four chickens for a pig, three pigs for two sheep, and five sheep for two cows, what is the minimum number of cows he needs to trade for $20$ chickens?

The minimum of cows he needs are: 2

Step-by-step explanation:

There's a relation between each animal:

5 chickens equals 1 pig

3 pigs equals 2 sheep

5 sheep equals 2 cows

You can understand it as the following three abstractions:

5c = 1p              (1)

3p = 2s             (2)

5s = 2o             (3)

Where:

c is for chickens

p is for pigs

s is for sheep

o is for cows

So now you have three equations with 4 variables. The next step is to obtain an equation that relates directly the variable c (chickens) with the variable o (cows). In order to do that from the equation 2 we obtain s in terms of p, as follow:

Then we replace s in the equation 3 and we obtain v in terms of p:

Now we replace v in the equation 1:

                   (4)

The equation 4 means that  1 chicken equals the fifteenth part of a cow. For this case the farmer needs 20 chikens, so we multiply per 20 each part of the equation 4:

As it is impossible to have 1.3333 cows, the answer  is 2 cows approximately.

d. it is a convex pentagon because it has five sides and none of the sides would extend into the inside of the polygon.

step-by-step explanation: