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Pasture Puzzles (Posted on 2013-11-01) Difficulty: 3 of 5

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.

See The Solution Submitted by broll    
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Solution Solution | Comment 1 of 10

a) If rate of grazing of cow is x and the rate of growth of grass is y and initial amt of grass is c and a and b is the time taken by 54 cows and 174 cows respectively to graze then

14x X 5a=c+y X 5a and 54x X a=c+y X a or 4x=y and c=50ax

38x X 5b=c+y X 5b and 174x X b=c+y X b or c=170bx or a/b=17/5

let t be number of cows that can completely graze for one day so

tx=c+y or 40a+4 or 170b+4 these are all non prime so if we consider 40a+3 or 170b+3 then the numbers are 1363,5443 ...etc where a and b are positive integers namely 34/10 and 136/40 respectively. Largest prime???

b) Was recently asked in another form. Sometime during the 13th day.

  Posted by Salil on 2013-11-05 23:44:27
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