I’ve been having fun solving puzzles from a really well-done and creative book by J. Ward, called “Alice’s Puzzles Through the Looking Glass.”
I’ve been able to solve each one after a careful reading of the puzzle, using a little thought, logic and simple arithmetic.
However, this one puzzle is driving me nuts! Here it is in a nutshell:
*Four black and three brown cows gave as much milk in five days as three black cows and five brown cows gave in four days.
Which kind of cow was the better milker?
*
I know the answer because I looked it up in the back, but it’s only one sentence stating whether it was black cows or brown cows which were the better milkers (I’ll reveal that a little later, if anyone’s interested). They don’t explain why. Even knowing the answer, I couldn’t work out how you’d arrive at that conclusion.
My question is, how the heck do you figure it out? Please keep in mind that I’m okay at arithmetic but I don’t know anything about higher math except a tiny bit of algebra. I don’t think it would involve any tough math, though, because none of the other puzzles do.
So could anyone explain this to me very simply, please?
Feeling very stupid 
It’s easiest with a little algebra.
Say that:
k is the amount of milk a black cow produces in one day
n is the amount of milk a brown cow produces in one day
We want to be thinking in terms of “cow-days”. This is just number of cows times number of days. 20 cows working for one day is the same as 4 cows working for 5 days, which is the same as one cow working for 20 days.
We have a given amount of milk and two different ways of producing it. One way is that we have 20 black cow-days (4 cows times 5 days) plus 15 brown cow-days. And the other way is that we have 12 black cow-days plus 20 brown cow-days.
To get the amount of milk, we have to multiply by the variables above. So our formula is:
20k + 15n = 12k + 20n
That easily simplifies to:
8k = 5n
And then:
k = (5/8)*n
k was the rate of black cow production, and we see that it’s only 5/8ths of brown cow production. To the brown cows are better.
It looks like a pretty simply algebra problem to me.
Black cows = x
Brown cows = y
The first herd is 4x+3y.
The second herd is 3x+5y.
We can also use these formulas to assign the daily milk production for the next stage.
You’re aiming for a quantity of milk, z. Herd 1 does it in 5 days, so z=5(4x+3y). Herd 2 does it in 4 days, so z also = 4(3x+5y).
Thus:
5(4x+3y)=4(3x+5y)
Which we can then simplify, thusly…
20x+15y=12x+20y
8x=5y
(8/5)x=y
Therefor, the brown cows (y) are the better milkers, giving 1.6 times as much milk as the black cows.
Thank you both very much! This proves I am truly dumber than I realized … I did get to the 4x+3y=3x+5y part but had no idea what to do next. I read both of your explanations a few times and think I maybe would be able to work out a similar problem next time, so I’m grateful that you took the time to explain. 
There go my illusions of having somewhat decent common reckoning abilities. I guess that’s what you get for taking one term of grade 8 algebra almost half a century ago. :o