7 Cows and 8 Chickens
Step-by-step explanation:
Note that cows have four legs each and chickens have two legs each.
if there were only cows, then 15 cows times 4 legs for each cow would give 15 x 4 = 60 legs. 60 is greater than 44, so some (maybe all) of the animals are chickens.
If there were only chickens, then 15 chickens times 2 legs for each chicken would give 15 x 2 = 30 legs. 30 is less than 44, so we have confirmed that there is a mixture of both cows and chickens on the farm.
One approach would be to build a table of possibilities:
Total legs will equal number of cows times 4 plus number or chickens times 2 and we know the total number of animals is 15, so we keep cows plus chickens equal to 15 and just keep adding a cow and subtracting a chicken and do the math until.we get 44 legs. We know that it will take less than 13 tries, since we already ruled out 15 cows or 15 chickens. We'll start with 1 cow and 14 chickens and go from there
1 Cow + 14 Chickens = (1 x 4) + (14 x 2) = 4 + 28 = 32 legs
2 Cows + 13 Chickens = (2 x 4) + (13 x 2) = 8 + 26 = 34 legs
3 Cows + 12 Chickens = (3 x 4) + (12 x 2) = 12 + 24 = 36 legs
4 Cows + 11 Chickens = (4 x 4) + (11 x 2) = 16 + 22 = 38 legs
5 Cows + 10 Chickens = (5 x 4) + (10 x 2) = 20 + 20 = 40 legs
6 Cows + 9 Chickens = (6 x 4) + (9 x 2) = 24 + 18 = 42 legs
7 Cows + 8 Chickens = (7 x 4) + (8 x 2) = 28 + 16 = 44 legs
We can stop here because we can see that 7 Cows and 8 Chickens have a total of 44 legs.
Another way to approach this is to use algebra and to make is shorter to write we can assign a variable to each item and relate them in some way.
Let's use the letter x to stand for cows and the letter y to stand for chickens so we don't have to keep writing chickens and cows. x and y are just common letters to use, by the way, could use w and k perhaps since cow has a w and chicken has a k...sadly they both start with c, so we can't use that for both.
We know a couple different, but related, things.
One thing we know is that the total number of animals is 15. We can write that as:
x + y = 15
That means cows (x) plus chickens (y) equals 15.
A second thing we know is that the total number of legs is 44. We can write that as
4x + 2y = 44
That means 4 times x (cow) plus 2 times y (chicken) is equal to 44 since each cow has 4 legs and each chicken has 2 legs.
Since we now have two related equations and two variables we can solve for the variables.
If we take the first equation and rearrange it, by subtracting x from both sides, we can see that one way to do it is the following
y = 15 - x
That means the number of chickens (y) is equal to 15 minus the number of cows (x). As long as we do the same thing on both sides of the equals sign the equation is still.equal...you will see this again below.
Now we can then substitute 15 - x for y in the second equation and then all we have is x in that equation. (We can do this because y is equal to 15 minus x...they both have the same value and mean the same thing...they are equal to each other)
4x + 2(15 - x) = 44
As you can see, we now have 2 times 15 minus x where we had 2 times y before. If we multiply the 2 through we get because 2 times 15 is 30 and 2 times x is 2x
4x + 30 - 2x = 44
We can subtract the 2x from 4x and we get
2x + 30 = 44
Now we can subtract 30 from both sides of the equation because as long as we do the same thing on both sides of the equals sign the equality holds true
2x + 30 - 30 = 44 - 30
30 minus 30 is zero and 44 minus 30 is 14 so now we have
2x = 14
If we divide both sides by two, we still maintain the equality and we get x on one side as 2x divided by 2 is 1x or just x and we get 7 on the other side as 14 divided by 2 is 7
x = 7
That means we have 7 Cows as x represents cows, so half the problem solved
If we now go back to our first equation,
x + y = 15
We can replace x with 7 and get
7 + y = 15
We can subtract 7 from both sides and maintain the equality and isolate y
7 - 7 + y = 15 - 7
Again, 7 minus 7 is zero so we have isolated y and 15 minus 7 is 8
y = 8
This means we have 8 Chickens since y represents chickens
Once again we have shown that Old MacDonald has 7 Cows and 8 Chickens
