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100 Herbivores (Posted on 2002-06-10) Difficulty: 2 of 5
There are a total of 100 animals: cows, sheep and buffaloes. These 100 animals ate 100 bunches of grass.

Every cow ate 5 bunches, every buffalo ate 3 bunches and every sheep ate only 1/3 bunch.

How many cow, sheep and buffalo are there? You only know that there is at least one of every kind of animal.

See The Solution Submitted by vohonam    
Rating: 2.5455 (11 votes)

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Solution Puzzle Solution | Comment 8 of 9 |

Let the respective number of cows, buffaloes and sheep be a, b and c, giving:
a + b + c = 100
5a+ 3b + c/3 = 100

Or, 2b + 14c/3 = 400
0r, 3b + 7c = 600
Reducing both sides in mod 3;
7c = 0 (mod 3); so that c must be divisible by 3 as 3 and 7 are coprime to one another
Let c = 3d. This yields:
b = 200 - 7d
So, a+ b+ c = 100 gives:
a = 4d - 100

Since there must be at least one kind for wach, it follows that
each of a and b must be positive.
Now b> 0 gives d<200/7< 203/7 = 29
and a>0, gives d> 25

So d can assume values between 26 and 28 inclusively.

 d = 26, gives (a, b, c) = (4, 18, 78)
 d= 27, gives (a, b, c) = (8, 11, 81)
 d = 28, gives (a, b, c) = (12, 4, 84)

So, (#cows, # buffaloes, # sheep) = (4, 18, 78); (8, 11, 81) ; (12, 4, 84)

 

 

Edited on March 24, 2007, 12:06 pm
  Posted by K Sengupta on 2007-03-08 12:08:00

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