Help for my daughter (mathematics related)

[Trillian]

Okay this is going to be a bit odd, but I have a question related to my daughters Grade 4 mathematics (yes, Grade 4 and I don't know the formula).

The question goes like this :

A farm has some chickens and some cows. Altogether they have 72 eyes and 92 legs. How many cows are on the farm?

Now I know the answer is 10 cows. But I worked this out through a process of elimination rather than a mathematical formula. I need to know if there is a formula for calculating this answer (and what that formula is) or if a process of elimination is the only way?

Thanks!

[Bluefinger]

Wouldn't you be looking for a formula that similar to this:

0 = y(2a+b) + x(a+b)

y for cows, x for chickens, a for legs, b for eyes.

Correct if I'm wrong, but that's what immediately popped into mind after a quick bit of writing down.

[catanon]

Actually, I think that works. I'll have to remember that little formula.

Advanced Math is being a bitch.

[purimpopoie]

Assuming that you're right about the 10 Cows, there are 26 Chickens.

EDIT: Oh, you don't need the number of chickens? Nevermind then. >.>

[Trillian]

Oct 7, 2009 6:17:17 GMT -5 Bluefinger said:

Wouldn't you be looking for a formula that similar to this:

0 = y(2a+b) + x(a+b)

y for cows, x for chickens, a for legs, b for eyes.

Correct if I'm wrong, but that's what immediately popped into mind after a quick bit of writing down.

Thanks Bluefinger, sorry for asking this, but I tried to work it out using that formula on my own, but I can't. Keeping in mind that both y and x are unknown, and only a and b are known, how would you get to the answer (i.e. 10)? Step by step, please?

I feel terribly stupid, since this is a fourth grade question!!

[m52nickerson]

x = cows
y = chickens

x+y=36 and 4x+2y=92

x+y-36=4x+2y-92

x+y=4x+2y-56

0=3x+y-56

3x=-y+56

x=(-y+56)/3

[Trillian]

Oct 7, 2009 7:47:15 GMT -5 m52nickerson said:

x = cows
y = chickens

x+y=36 and 4x+2y=92

x+y-36=4x+2y-92

x+y=4x+2y-56

0=3x+y-56

3x=-y+56

x=(-y+56)/3

:-)
I actually did understand that, but I'm afraid if she gave that answer in class, they would probably ship her off to some gifted school or something.

But thanks though, the more I think about it, it seems the process of elimination is the only way to get to the answer (10), because you have two unknowns, so mathematically, your answer is always going to be an "xy" combination. But at this level they are looking for a numerical answer.

[Bluefinger]

Oct 7, 2009 7:44:12 GMT -5 Trillian said:

Oct 7, 2009 6:17:17 GMT -5 Bluefinger said:

Wouldn't you be looking for a formula that similar to this:

0 = y(2a+b) + x(a+b)

y for cows, x for chickens, a for legs, b for eyes.

Correct if I'm wrong, but that's what immediately popped into mind after a quick bit of writing down.

Thanks Bluefinger, sorry for asking this, but I tried to work it out using that formula on my own, but I can't. Keeping in mind that both y and x are unknown, and only a and b are known, how would you get to the answer (i.e. 10)? Step by step, please?

I feel terribly stupid, since this is a fourth grade question!!

Ahh, basically, the formula I put down is simply the relationship of the whole arrangement of cows to chickens with regards to eye to leg ratio. Chickens have a pair of legs and a pair of eyes (1:1), whilst cows have four legs and a pair of eyes (2:1). We know the full numbers of eyes and legs, but not of either cow and chicken, therefore, we subtract the number of eyes from the number of legs in order to get the value of the difference between the number of eyes to legs.

Example: 24 - 18 = 6

Since cows have twice as many legs as they do eyes, we just divide the difference value by two to get the answer, which in my example would be three. If it were only chickens in the group, we'd have the same number of eyes to legs, so we can assume that any difference is due to there being cows in the group. Does this help?

[m52nickerson]

Oct 7, 2009 7:56:38 GMT -5 Trillian said:

I actually did understand that, but I'm afraid if she gave that answer in class, they would probably ship her off to some gifted school or something.

But thanks though, the more I think about it, it seems the process of elimination is the only way to get to the answer (10), because you have two unknowns, so mathematically, your answer is always going to be an "xy" combination. But at this level they are looking for a numerical answer.

I think your right, seeing that at 4th grade I doubt they at graphing yet.

[Mantorok]

Every animal has a minimum of 2 legs. Subtract 72 legs from 92 legs to get 20 extra legs. Cows have 2 more legs than chickens, so divide the 20 legs by 2 to get 10 cows.

[Trillian]

Thank you all very much.
Bluefinger and Mantorok, you both said pretty much the same thing and it seems so easy now that I know the answer (doesn't it always)!!
At least now my poor little eight year old doesn't have to deal with my "round and round" explanation and I can explain it to her in a way she'll understand!!

[davidh72000]

Oct 7, 2009 7:47:15 GMT -5 m52nickerson said:

x = cows
y = chickens

x+y=36 and 4x+2y=92

y=36-x

Now substitute into 4x+2y=92 (or 2x+y=46).

2x+(36-x)=46
x+36=46
x=10

10+y=36
y=26

So there are 10 cows and 26 chickens.

Edit:
I should also mention that the formula given in the first reply is not correct.

[m52nickerson]

Oct 7, 2009 12:22:30 GMT -5 davidh72000 said:

Oct 7, 2009 7:47:15 GMT -5 m52nickerson said:

x = cows
y = chickens

x+y=36 and 4x+2y=92

y=36-x

Now substitute into 4x+2y=92 (or 2x+y=46).

2x+(36-x)=46
x+36=46
x=10

10+y=36
y=26

So there are 10 cows and 26 chickens.

Edit:
I should also mention that the formula given in the first reply is not correct.

Yes! I realized this after I had posted. You really don't have two variables because you can use their relationship to define one or the other.

I did it this way......

x + y = 36

y = 36 - x

Then....

4x + 2y = 92

4x + 2(36 - x) = 92

4x + 72 - 2x = 92

2x + 72 = 92

2x = 20

x = 10

[Old Viking]

The proper answer to "How many cows are there on the farm?" is "Ask your mother."

[Trillian]

Thanks David and M52.

Yeah, Old Viking, that would work if mom knew how to answer the question ;-)

Glad I could come here for help though!!