See Slow Cows

(a) More Slow Cows.

If 14 cows take exactly 5 times as long to graze out a pasture as 54 cows, and 38 cows take exactly 5 times as long as 174 cows, what is the largest prime number of cows that can graze the same pasture?

Note 1: In this part, all times are calculated in exact whole numbers of days, and the pasture is of the smallest size compliant with the given conditions.

(b) Newton's Goats

Six goats eat all the grass in a pasture in exactly three days, while 3 goats take exactly seven days to graze out the same pasture. On what day will the pasture become exhausted, if two goats are grazing there?

Note 2: In both parts, the grass grows at a steady rate, and each animal eats the same, constant, amount each day; but, unlike cows, goats eat grass down to the stubble, so that areas that have already been grazed do not resume growth while the pasture remains in use.

(b) The answer remains the same 13th day

If x is the rate at which the goat eats and y is the rate of growth of pasture and c be the initial grass in the pasture and a is the time taken by 2 goats to finish grazing the pasture then.

6x X 3 = c+y/3(1+2+3) or 18x=c+2y

3x X 7 = c+y/7(1+2...+7) or 21x=c+4y

so y=3/2x and c=15x

2x X a=c+1/a(1+2+...+a)y or a=63/5 =12.6 days

Hope I understood it correctly.