This is actually somewhat complex. You need to take into account the fact that, while the grass is being eaten, it is also growing. At some point, the fields are grazed to the point of uselessness, but it isn't just a matter of the grass being eaten; you also have to account for regrowth.
From the first field, we get that 21 horses take 9 weeks to graze the 10 acres down to the ground. During that time, the grass is growing at some rate G cubic centimeters per week per acre. (It's "cubic" because we're taking the area of an acre and multiplying by Since the fields are assumed to start from the same conditions, this means that we have 10 acres that start at some volume, based on the original height, of H cc/acre of grass. The horses eat this at some rate E cc per horse per week. So, after nine weeks at this rate, we have. . . . .
(10 acres)(H cc/acre) + (9 weeks)[(10 acres)(G cc/acre-week). . . . . . . .
- (21 horses)(E cc/horse-week)] = 10H + 90G - 189E = 0
In the same way, the second field gives us:. . . . .
(10/3 acres)(H cc/acre) + (4 weeks)[(10/3 acres)(G cc/acre-week) . . . . . . . .
- (12 horses)(E cc/horse-week)] = (10/3)H + (40/3)G - 48E = 0
Then what? We have two equations, but with three variables:. . . . .
10H + 90G - 189E = 0. . . . .
(10/3)H + (40/3)G - 48E = 0
We know that we'll have 24 acres for the third field, and horses will be grazing there for 18 weeks. So we know that the third equation will be:. . . . .
24H + 432G - (x horses)(18E) = 0
If we solve the first two equations for "10H=", we can get a ratio of values:. . . . .
10H = 198E - 90G. . . . .
10H = 144E - 40G
Then:. . . . .
198E - 90G = 144E - 40G. . . . .
54E = 50G. . . . .
27E = 25G. . . . .
E = (25/27)G
Then:. . . . .
10H = (144)(25/27)G - 40G. . . . .
10H = (400/3)G - (120/3)G = (280/3)G. . . . .
H = (28/3)G
Does this help? Let's see:. . . . .
24H + 432G - (x horses)(18E) = 0. . . . .
24H + 432G - 18x(25/27)G = 0. . . . .
(24)(28/3)G + 432G - (18x)(50/3)G = 0. . . . .
200G + 432G - 300Gx = 0. . . . .
632G - 300Gx = 0
You can solve this for "x=", since the two Gs will cancel out.
Check my math, though; because it seems reasonable that you should
end up with a whole number. I'll be back later to proof the above....